BYM:
Firstly, as I’m sure you have already discovered in your research of the topic, there is a halachic distinction between one who teaches a woman Gemara and a motivated woman who wishes to learn it on her own. No less a personage than the Chida writes (in Tuv Ayin) that the original prohibition cited by R’ Eliezer does not even apply in the case of a woman who is truly MOTIVATED to learn Gemara. Obviously, there are further voices of agreement and disagreement on the subject, but I am fairly confident you have studied these, too.
I guess my question to you is: which perceived fallacies and inconsistencies PRECISELY do you mean to resolve through learning Gemara? Rigorous study of Gemara is, as you theorized, rooted strongly in building sound logical edifices and constructing sound arguments. On a purely logical level, it is immensely satisfying to a person with a scientific mind. But here is where I must digress.
Without the basic tenets of belief intact, you may as well be having an in-depth discussion of the ins and outs of Tolkien’s Middle Earth or the Harry Potter series. In other words, you can have an immensely satisfying discussion about something that you are taking for granted to be fictitious. The discussion will be enjoyable, but the MEANING will be absent.
I myself have had a long and interesting journey of belief, and I won’t even be so arrogant as to say that it’s ended. I have learned in my travels that Talmudic Judaism’s logical structure stands on a base of spiritual BELIEF. Why discuss whether or not a divorce is valid if it is handed to one’s wife via a messenger if you don’t believe in the binding nature of that document? Why debate the hidden meanings of extra words or letters in the Torah if you do not believe them to be of divine origin?
My question to you is: what exactly is it that you seek in the study of Talmud? Is it the satisfaction of logical construction or is it the foundation of belief itself?