They will be allowing transgender people to join next summer. There is a great Facebook group for people that are interested in this. If you would like to be added, feel free to PM me once you are able to and I can connect you to the admins there.
I'm not sure how the new open service policy for trans people will affect Selecive Service. Since I frequently work for the government, I had to file for a Selective Service exemption saying that I was transsexual. I'm not sure what their plan for this is, so I would hold off on registering for Selective Service unless someone here has current info on that.
As far as your concerns:
Before you join, you would need all of your identification to match if possible. So if you were born in a state that does birth certificate amendments, you would need to have that done. If that's not the case where you were born, you may just be able to tell them that. Or they may want something notarized saying as much.
You do need to have documented RLE as male. I assume a letter from the doctor that does your HRT will suffice, combined with updated documentation. AFAIK, you are not required to have any surgeries pre-enlistment, but you may be more comfortable if you do. I am considering joining after my bottom surgery is over with.
They would be getting access to all of your history - personal, education, medical, financial, etc. I can't say for sure what is disqualifying and what isn't in terms of personal history. And what may disqualify you from one branch may not disqualify you from another. I have a friend with bad credit who is having issues enlisting in the Air Force, but the Army has offered to take him if he can't get it worked out.
Sexual assault and harassment are crimes, and anyone doing those things would be punished under military law. So while I can't say that those things would definitely never happen to you, there is at least recourse if they do. As far as punishment, I'm assuming you mean punishment for being trans. If that is the case, then the above applies. If you just mean punished in general, I would read up on boot camp and military training. A lot of military team building relies on what some may view as punishment.
You could do any job you qualify for. ASVAB scores determine what you're eligible to be. The one thing you'd want to keep in mind is whether jobs are in demand or not, and how that effects your timeline for entry. If jobs aren't in demand you may not be scheduled for basic training right away.
Like Deborah said, your best bet is to speak with a recruiter for the branch you want to go into. They would be able to speak more about the specific requirements you'd need to meet prior to joining. And like I mentioned above, I'd be happy to connect you to a peer support/mentorship group on Facebook if you're interested.