Examples of the above arguments

Argument 1: No significant relationship between environmental/physical similarity and phenotypic similarity in twins
"Measures of the degree of similarity in parental treatment turn out to not be correlated with similarity in IQ or other personality measures."\cite{Cesarini_2009}, (p. 828)
"...mothers' and fathers' approach to raising twins had no significant influence on twin resemblance for the four examined psychiatric disorders [namely, major depression, generalized anxiety disorder, phobia, and alcoholism]. These results suggest that the differential treatment of MZ and DZ twins by their parents is unlikely to represent a significant bias in twin studies of these major psychiatric disorders."\cite{Kendler_1994}
"If the greater genetic relatedness of MZ twins was irrelevant to their observed greater ideological similarity, then MZ and DZ twins should display substantially reduced correlations when environmental similarity or contact is reduced. In addition, the correlations for MZ and DZ pairs should be approximately equal across levels of environmental similarity and contact...this is not the case."\cite{Smith_2011}
"...in the relatively rare cases where parents miscategorize their twins as MZ instead of DZ (or the converse), differences in correlations of cognitive ability and personality persist."\cite{CESARINI_2010}, page 1742
"Controlling for environmental similarity reduced heritability significantly for only one out of the 32 outcomes examined: neuroticism."\cite{Felson_2014}
"The EEA was investigated with a linear regression model, examining if the twins contact frequency predicted within-pair differences in PA [physical activity], and further by a simulation study. We found no support for violation of the EEA."\cite{Eriksson_2006}
"...the validity of the equal environments assumption has been evaluated using mislabeled twins (twins labeled DZ when they are in fact MZ) or and MZ twins who are in fact treated differently. Both methods rely on the idea that MZ twins who are treated more individually should show more differences than those who are treated more similarly. Studies using both methods provide evidence for the validity of the assumption."\cite{Kohler2011}
"...studies that examined twin physical similarity or parental behavior towards twins as well as the effects of perceived zygosity (where DZ twins were identified incorrectly as MZ)...have concluded largely that previous assertions of violations of EEA may have been overstated."\cite{Agrawal_2008}
"...similarity of friends or parental treatment is not associated with twin similarity in personality, interests, or abilities."\cite{Bouchard_2002}
Argument 2: Greater genetic similarity of MZ than DZ twins causes their greater environmental similarity
For a list of examples of this argument, see \cite{Joseph_2012}.
Argument 3:
"...findings garnered from studies using the classical twin design, the adoption design, and the twins reared apart design largely converge"\cite{BARNES_2014}
"...the results of adoption studies and studies of twins reared apart are generally consistent with the results of classical twin studies."\cite{Felson_2014}
Criticism 2: But they aren't identical!
Twin studies assume that MZ twins are 100% genetically identical, which is not true. For one, they have significant differences in their epigenetic profiles.\cite{pharmacy}\cite{Fraga_2005}\cite{Haque2009}\cite{Petronis2003} Yet another classical twin study assumption bites the dust. Or does it? Twin researchers can reply to this claim in a number of ways, but their preferred approach is to say that there are no better ways to control for genetic factors than by comparing MZ and DZ twins, because they are so close to being 100% and 50% identical, respectively.
For instance, Verhulst & Hatemi (2013) argue that "... there is no better control of gene expression than identical twins, and there are no better controls for both family environments and genotype than identical twins. Indeed, the best models of genetic expression for complex human traits involve identical twins."\cite{Verhulst_2013} Similarly, Segal et al. (2011), in their response to a Slate article slamming classical twin studies, describe genetic and other biological differences between each twin in a pair of MZ twins. They then argue that "Despite these biological differences, MZ twins (whether reared apart or together) are more alike than any other pair of individuals. Moreover, to the extent that MZ twins are not genetically identical, the true importance of genes is actually underestimated by the MZ-DZ comparison (my emphasis)."\cite{Segal_2011} 
This argument (that identical twins not being identical makes heritability estimates from twin studies too low, not too high) is also a common one among twin study defenders, ranging from Segal et al. to Tabarrok's and Collins' responses to the same 2011 Slate article.\cite{revolution}\cite{heritabilitya} It has received support from a recent study in Behavior Genetics, which found that if identical twins aren't quite identical, this leads to heritability estimates being higher than, and only slightly different from, what they should be, based on mathematical simulations.\cite{Liu_2017} Of course, whether these simulations accurately represent actual biological or epigenetic processes is another matter entirely.
Criticism 3: What about epigenetics and gene-environment interactions?
A recent paper by Richard Lerner and Willis Overton, both longtime and outspoken critics of behavior genetics, argues that "epigenetics invalidates all models involving genetic reduction". They argue that this is the case because, contrary to genetic reductionist models such as that of the classical twin study, "Genes do not determine behavior" (Lerner & Overton 2017). Note that such models are supposed to include behavior genetic models, which they are accusing of being "genetic reductionist" because they (supposedly) grossly oversimplify the ways that genes affect human traits. Another recent paper reviewed the literature on the genetic contribution to aging traits and concluded: "...it is likely that epigenetics contributes to heritability because MZ twins (particularly dichorionic who have split earlier in development) have less DNA methylation difference than DZ twins."\cite{Steves_2012}
And then there is the issue of gene-environment interactions (often abbreviated GxE). As a 2004 Monitor on Psychology article noted, "Some researchers think that interactions between genes and environment, rather than genes and environment separately, may influence many traits. A recent study from Science (Vol. 297, No. 5582) by Avshalom Caspi, PhD, of King's College London, for example, suggests that a gene might moderate propensity for violence, particularly in people who are severely maltreated as children. Many twin study designs don't take this type of complication into account" (Winerman 2004).
But wait, you haven't heard behavior geneticists defend their beloved twin studies against these accusations yet. So what are their rebuttals?
You could take the approach of Barnes et al. (2014), who simply write that "...much of the GxE literature is under heavy criticism and epigenetics is in its infancy". They also use the argument that many (perhaps 90%!) of the findings of reported GxE interactions are false positives to cast doubt on whether such findings exist to a significant extent. 
Alternatively, rather than questioning whether GxE effects occur, you could acknowledge that they do, but then say that behavior geneticists are already well aware of them and incorporate them into their models regularly. You could also argue that, sure, the CTD is seriously flawed because it ignores GxE interactions, but argue that behavior geneticists have other types of models they can use that do take into account such interactions. Consider the following quote from a response by two "genopolitics" researchers to a critique of the CTD: 
"GE interplay is clearly ingrained within the theory and statistical model of modern twin analyses... the fact that the CTD does not estimate GE interplay is not a failure of the model. If GE interplay is suspected, then extensions of the CTD are available and should be utilized" (Verhulst & Hatemi 2013).
After citing this paper, Hatemi et al. (2014) argue that "Behavior genetic approaches have also inspired philosophical objections due to the firm belief that behavioral differences are entirely socialized. Several publications in the political science literature have begun to erode this view and explicate the theoretical and empirical justification for inclusion of genetic influences and biological mechanisms in general, including gene-environment interplay"\cite{Hatemi_2014}.
So what about epigenetics? Do BGists have any decent responses to the "epigenetics refutes the CTD" argument? Well, they could say this: "Most researchers consider political traits to be influenced by thousands of genetic markers both indirectly and through interactions with numerous environmental stimuli and other genes in complex genomic, epigenetic, and neural pathways. By contrast, many criticisms are developed as if responding to the view that political traits are simple Mendelian traits, governed by a single gene or a small set of genes."
Another time-honored (but questionable) method is to do a simulation study estimating the effect of (be warned, this is a mouthful) differences in genetic similarity between MZ and/or DZ co-twins from assumed values on heritability estimates. In classical twin studies, genetic similarity between MZ and DZ co-twins is assumed to be 100% and 50%, respectively, and a recent simulation modeling study estimated that when the actual similarity is different from the assumed values, it leads to wrong estimates, but not that wrong: "Although estimates of genetic and nonshared environmental influences from the standard biometric model were found to deviate from “true” values, the bias was usually smaller than 10% points indicating that the interpretations of findings from previous twin studies are mostly correct."\cite{Liu2017}
In short, behavior geneticists argue that the evidence of GxE interactions and epigenetic effects is too preliminary and/or weak and/or inconsistent to cast much doubt on the results of twin studies. Unless they don't, in which case they say, "Uh, we already know about this shit, so you don't need to tell us, and we already take these phenomena into account", and "Don't attack a straw man, we don't think political beliefs are only determined by a single gene, of course other factors are important!"
Criticism 4: These studies assume random mating
Barnes et al. (2014) note that classical twin studies assume random (or assortative) mating. They state further that "...violation of the random mating assumption leads to inflated estimates of the shared environment effect and deflated estimates of heritability." Similarly, Verhulst & Hatemi (2013) write that "...CTD models assume random mating and due to the very large correlation between spouses (r ≈ 0.6) for attitudes, the CTD actually underestimates the magnitude of genetic influences".
Some researchers have used the "nuclear twin family model" to account for this assumption, and the assumption that dominant genetic effects and shared environment aren't present together, in the CTD. This has yielded somewhat lower estimates of heritability.\cite{Burt_2012}
Criticism 5: These studies assume MZ twins, DZ twins, and singletons all have the same prevalence for the trait under study
The CTD assumes that the trait being studied is equally prevalent in MZ twins, DZ twins, and singletons (this last word refers to people who aren't any kind of twin).\cite{Thomsen_2014} BGists respond by simply saying that this assumption is true, e.g.: "Twins are indeed representative of the general population for a wide range of traits and diseases..."\cite{Evans_2000} And in many other studies by BGists, the same claim has been made, namely that twin samples are representative of the general population: 
But other researchers are more cautious, for example: "Twins are representative of the general population for some but not all measures of ocular biometry. Consequently, care should be taken when extrapolating twin data for these traits in heritability and other genetic studies."\cite{Sanfilippo_2011}
Criticism 6: Like, what even is heritability, anyway?
Critics of BG have long noted that the concept of "heritability" (at least in the context of human BG) is highly misleading because it seems to mean "genetic determination", but it really does not mean that at all. Instead, it just means the % of variation in a population in a given trait that is due to genes. "Genetic determination" is an individual-level characteristic, not a population-level one like heritability, so heritability says nothing at all about how much of any trait YOU have is due to YOUR genes.\cite{Sauce2018} As Ned Block noted more than two decades ago, it is therefore meaningless to ask, for example, "What's the heritability of my IQ?"\cite{Block_1995} Also, heritability analyses, which are based on the analysis of variance (ANOVA) statistical technique, are not very good at detecting gene-environment interactions if they are present.\cite{Wahlsten_1990} Lastly, since the concept of heritability was invented in the 1940s for predicting the success of selective breeding in farm animals,\cite{Wahlsten_2003} it is very unlikely that all, or even most, of the assumptions on which heritability estimates for the purpose of selective breeding depend will be met in normal human populations. As Stoltenberg (1997, p. 91) noted, "...human populations are much more unlikely [than farm animals] to satisfy the host of assumptions that undergird partitioning behavioral variance into the P = G + E model. Such assumptions include no dominance deviation, no epistasis, no assortative mating, no genotype × environment (G x E) interaction, and no GE covariance".\cite{Stoltenberg_1997}