What Happens Now? And What The LGBTQ Can Do
1) Marriage rights
When gay marriage became legal nationwide, it wasn't because a new law was passed. It was because the Supreme Court determined that it was already unconstitutional to not allow gay people to get married. In order to remove federal gay marriage, the administration would need to draft a bill, pass it with a majority, and get it signed by Trump. It would then be instantly challenged by basically every left-wing legal body in the entire country, and it would make its way to the Supreme Court where...they would probably overturn the new bill. Because the Supreme Court rarely overturns it original decisions, they have to have "compelling and apparent" reason to undo something they've already done. This is unlikely to happen, even with a conservative Supreme Court.
Is it possible? Yes. But a lot of states would likely ignore the order on the grounds of states' rights, which is funny because this whole administration is supposed to be about returning power to the states. That's the general idea behind the parties. Liberals prefer a larger government with more control, and Republicans prefer a smaller government with less control. Of course, ironically people tend to become more repressed under a "smaller" government, but that's another story.
2) Anti-Discrimination Laws
This one is tougher. Technically Trump can put an executive order into his first day the undoes the ones Obama has done that protect LGBT people. Is this going to happen? The answer, surprisingly, may be "no". I didn't vote for Trump, because I would rather peel my skin off with a cheese grater made of salt, but I follow politics closely and so far Trump has been talking about just about everything except LGBT people.
He does actually seem largely disinterested in us. He's focused on the economy. His student-loan repayment plan is actually not insane, which made me do a quadruple-take when I read it.
That being said, Pence could easily poke him in the ribs until he's willing to sign executive orders like that into law. So, what happens then? Well, states' rights are still a thing, so if you're lucky enough to live in a state that outlawed LGBT discrimination before Obama did, you probably won't notice much change. If not, well...save your pennies and take a bus? I don't know. It's a tough situation, and hopefully Trump is worried about bigger stuff and refuses to hear about Pence's anti-LGBT obsession.
3) Bathrooms
Same as above, this is something that could be undone on day one if he cared enough to do it. He may or may not. States can determine for themselves what they want to do, and it's unlikely that any order to force the states to be like crazy North Carolina is very likely in a government that is always beating its chest about "small government" and "power to the people".
If it happens, trans people are going to have to be adaptable. I know of two or three apps that tell you where the gender-neutral restrooms are in your city. Or, if you pass well enough, I say go ahead and use the bathroom you want (especially if your gender is updated on your ID already).
Basically, at worst we're looking at dealing with what trans people dealt with fifteen years ago. It's bad, but it's not as bad as what people are making it out to be. You don't have to submit to a DNA test every time you need to pee. Just be smart about when and where you go, to the best of your ability.
4) Hate crime legislation
Even our current administration generally considers hate crimes a bad thing. Republicans, despite popular opinion around here, are not actually Satan-incarnate. They just have outdated views that are pretty bad for us LGBT types and other minorities.
There's no way the government is going to attempt any kind of legislation that makes hate crimes legal, or reduces the punishment. The ACLU, which has already promised to hound Trump from his first day to his last, would be over that so fast your head would spin. They'd cite hundreds of legal precedents and demonstrate why, no, you can't build a legally defensible position toward approving of hate crime.
Now, on some level, it's up to states to determine what they consider a hate crime and how severe the punishment should be, so living in more liberal states is still preferable here. But really, I don't expect this one to change much.
And, finally, you asked what can be done to fight for those rights. Here's my answer:
Be vigilant. Be aware. Do not let yourself get complacent even if Trump assumes office and nothing bad happens to us. Keep an eye out for news about laws and policies that would be damaging to our rights, and then say something. Post and link it on Facebook. Call the ACLU. Send a letter of complaint to your local representative. People think that these things don't matter, but they absolutely do. If enough people write in to complain about a policy that someone is trying to enact, it at least gives them a reason to consider not enacting it.
The other best thing you can do is not panic. I know this is incredibly scary, and I won't lie to you, there's a good chance we're going to lose a lot over the next four years. This is especially true with health coverage and legally being able to change names/gender markers. But you absolutely have to stay strong and keep reminding yourself that this is temporary. Politics is like a pendulum. They're going to come in and make a mess, and in two years when the midterms get here, Dems are going to flood into the senate and start cleaning up. In another two years, we're almost certainly going to get either a Democrat or, judging by how anti-establishment people seem to be (see: Trump), something else entirely.
Don't panic. Remain calm. Be rational in your thoughts and actions. More than half the country wanted Clinton in office, and public opinion about LGBT people is actually more favorable now than ever before. The average everyday person is supportive of LGBT rights. Some of the 26% that voted for Trump may not be, but they are hardly representative of the country overall.
Not every employer, school, restaurant, etc, is going to discriminate even if were perfectly legal. "You can't legislate morality", as they say. And, personally, I'd rather not work for a company that would love a chance to fire me over being trans anyway.
It's going to be a little stressful and a little bit of a balancing act, but definitely not the end of the world.
One last piece of advice: if you haven't updated your gender on your driver's license and passport, now is the time to do so. Right now you don't need your birth certificate to match your identified gender in order to get a passport that does, but the window for that is closing in January. This is probably the one place that we'll be the hardest hit.
TL;DR: This is important info, go back up and read it.
