Would the eightfold path help someone transcend gender dysphoria? Is gender dysphoria a matter of clinging or a matter of hardwired biology?
As already mentioned, any sort of identity matter is dealt with as anatta in buddhism as far as i can tell. That likely goes for any form of gender identity, however it may present itself.
However, i'm not sure that the idea of anatta pertains to transcendence, because the word seems to connotate tanha in my understanding: It may lead us into desiring the notion of an elevated state, whereas anatta is arguably a question of no state at all.
The second half of your question seems like a question deemed unanswerable in buddhism (avyākata). For instance, this is mentioned in the Sabbasava Sutta:
"This is how he attends inappropriately: 'Was I in the past? Was I not in the past? What was I in the past? How was I in the past? Having been what, what was I in the past? Shall I be in the future? Shall I not be in the future? What shall I be in the future? How shall I be in the future? Having been what, what shall I be in the future?' Or else he is inwardly perplexed about the immediate present: 'Am I? Am I not? What am I? How am I? Where has this being come from? Where is it bound?'
"As he attends inappropriately in this way, one of six kinds of view arises in him: The view I have a self arises in him as true & established, or the view I have no self... or the view It is precisely by means of self that I perceive self... or the view It is precisely by means of self that I perceive not-self... or the view It is precisely by means of not-self that I perceive self arises in him as true & established, or else he has a view like this: This very self of mine — the knower that is sensitive here & there to the ripening of good & bad actions — is the self of mine that is constant, everlasting, eternal, not subject to change, and will stay just as it is for eternity. This is called a thicket of views, a wilderness of views, a contortion of views, a writhing of views, a fetter of views. Bound by a fetter of views, the uninstructed run-of-the-mill person is not freed from birth, aging, & death, from sorrow, lamentation, pain, distress, & despair. He is not freed, I tell you, from suffering & stress.
The first step of the Satipatthana is to see how impure the body is. Bikkhu analyo has a guide on this https://www.buddhismuskunde.uni-hamburg.de/pdf/5-personen/analayo/satipatthanapracticeguide.pdf
Probably not. I mean, ultimately it would but for all practical purposes, this is something you'll have to contend with for the rest of your life. One of the great misconceptions about Buddhist practice is that enlightenment will somehow solve all of our personal, psychological issues. That's simply not the case. Enlightenment is not perfection; neither is it complete freedom from all of our obstacles and misconceptions. Only a fully realize Buddha - one without karmic remainder - is free from all personal difficulties.
There's a koan in the Zen tradition that's sometimes given those who have just experienced daikensho or "great awakening". It goes something like this: a bull passes through a window, but the tail remains. Why can't the tail pass through? Even when we wake up, we still have that little bit of tail that we have to reconcile. In fact, the greater part of our practice post enlightenment will be spent addressing it. While that tail ceases to be a form of personal anguish, it's still there asking for us to deal with it. Think of enlightenment like a dinner party. While you get to enjoy the joy of liberation, ultimately, those dishes are going to be waiting for you the following morning. Fortunately, the enlightened heart is one hell of scrub brush.
(And I can't take credit for that last line. It's a riff on Jack Kornfield's book After the Ecstasy, The Laundry.)