A transsexual woman is:
a male-to-female (MTF) transsexual or transgender person and the term trans woman is preferred by some individuals over various medical terms.
Because a transsexual woman would view herself as of the female gender (though of the male sex), the term "gay" applies as it would to any other woman. That is, a gay transsexual woman is a gay woman, who happened to be born as a man.
As for your original questions --
Is it a man who became a woman, then turned gay and started liking other women?
Or is it a woman who likes other women (thus is gay), and then turned into a man?
You cannot, simply by the English language, tell which came first. Sexuality and gender identity are fluid and personal, and standard phrases (such as "gay transsexual woman") will not be able to tell you whether the individual first realized that they were gay, or first realized that they gender-identified as a woman. Only the individual, if they feel comfortable telling you this, will be able to explain.
What if you changed the word order, and got a "transsexual gay woman"? Then breaking it down, you have a gay woman. This woman is also a transsexual. But you don't know which part came first still--were they a transsexual woman who later realized she was gay? Maybe. Perhaps they just feel more comfortable identifying as a gay woman, who is transsexual. This is their identity, but the term does not capture which came first.
There is no universal affix in English equivalent to the German suffix behaftet.
Instead, there are two different ways to form this kind of adjective in English.
The suffix -ful suggests that the noun modified has the quality in question in abundance. It is a limited suffix that can be used only with certain words: hopeful, but not hungerful.
The past participle of some verbs can also be used as an adjective. For example, the opposite of "stainless" is "stained." Again, this does not work with all verbs, and I'm not sure what the rule is. It seems to be more common with transitive verbs ("stained," "baked") but also includes some intransitive verbs ("fallen," "wilted"). It cannot be used with nouns, but is often used with words that share a noun and verb--all four of the examples above fall into this category, and it may in fact be a requirement.
Your specific examples don't fall into either category. They're both nouns, so the verb tense trick doesn't work. ("Dimensioned" is a word but suggests "having been dimensioned" rather than "having a dimension.") And neither is on the -ful list.
So you have to use a multi-word construction instead of a suffix. This is pretty common in translating from German, which, from a native English speaker's perspective, seems to have an affix for just about everything.
Best Answer
Yes, nouns can have opposites. For example, the North is the opposite of the South, at least in the American Civil War, just as left and right are opposite in direction — and in theory opposite in politics.
The problem is that most nouns can be thought of as having many different properties, and you can easy flip a property on a different axis than the one that people are thinking of.
There are many other possible axes you can flip, all of which are the “opposite” of man along that axis.
Other examples of noun pairs that most people would think of as being opposite each other include: