How can I appropriately handle cleaning of gender data?
There are at least two general considerations to make:

Domain-related

If an attribute potentially has predictive power in your domain and more specifically for your task your models might benefit from a direct encoding. For example: if being trans is correlated with different psychological disorders then I'd include a direct feature for this. This way it is easier for your model to make a prediction since it does not need to combine two features in the first place (e.g. no need to combine "sex at birth" and "gender identification" to identify a transsexual person (which would not even accurate since "trans" is a much broader term than just sex at birth != gender identity)). 

Moreover, I'd apply the same thinking to other feature engineering questions. Sex has predictive power for many tasks related to mental disorders, e.g. because mood disorders are more common among women and anti-social personality disorders are more common among men. However, whether these are rather related to the sex at birth or the gender a person identifies with is another question. So if your hypothesis is that in your task the gender a person identifies with is important then, again, it makes sense to include this in addition to the sex at birth.

Model-related

Different models are able to handle predictors differently. For example, tree-based models can more easily work with two separate attributes sex == female and trans == True to implicitly derive trans female == True. However, linear models like neural networks might benefit from having a combined binary feature female trans.