Hey 92,
I enlisted in 2004 and got picked up STA-21 while in nuke prototype training 1-2 years later. I'll try to answer some of your questions with my personal experience, observations and opinions:
To get accepted for nuke STA-21 you need good grades in nuke training. That's mostly what it boils down to. They grade on a 4.00 scale for both A-school and Power School. While A-school does matter, your power school grades are probably the most important factor. The class average is usually somewhere between 2.9 and 3.3. In order to have a borderline chance, you need a 3.60. 3.70-3.79 and you've got a solid chance of getting picked up. Close to 3.80 and higher is almost a lock to get picked up. So, in essence, you need to be a rock star student near the top of your class which is hard to do. This is just for Nuke Officer STA-21 (carriers and subs). I'm not sure if the numbers are the same, but most years they'll accept 50 nuke STA-21's and typically 30-35 are students, and 15-20 staff... so if you can manage to be top 5 out of the 25-30 in your section, you've got a decent chance to be once of those 30-35 getting accepted that year.
To realistically get picked up STA-21 Core (you can pick whichever Officer job you want essentially) you're competing against the whole fleet, not just nuke staff and students, so you need to put some time in on a boat and get some great evals for a couple of years before having much of a shot to get accepted STA-21 Core and leave the nuke community. As far as reapplying, you can apply every year and getting denied one year shouldn't hurt you when applying for it the next year. They may even admire you a bit more for your determination and resolive.
Another option that I think might be even better for you is the Naval Academy. You can get picked up for the Naval Academy out of Nuke school, but if I remember correctly, you can't apply for both that and STA-21 at the same time. Since you are very young and have no college experience, I think the Naval Academy might be a great option for you to persue. Getting your degree as enlisted and going OCS while technically possible, isn't really feasible IMHO.
As far as what an officer does... you don't really have a lot of options. If you go enlisted to officer from the nuke program, they will in all likelihood force you to go subs. They said they'd never force me to go subs, but that's what they ended up doing, and what they've done for the past 3 years now due to being undermanned there. For STA-21 you will get sent from nuke training to NSI in Newport, RI for a mini-bootcamp like experience, then report to one of the 19(?) colleges with approved programs for nuke STA-21. You'll then have 3 years to complete your degree there while receiving full pay and allowances and finally get commissioned an Ensign. If you go Naval Academy, you'll probably be sent to NAPS in Newport, RI first for 6 months or so to do some academic prep work, then go to the Naval Academy afterwards and commission.l
After that, you'll go back to power school (a lot harder the second time around for me) and prototype, then to SOBC (learn sub tactical stuff) for 2 months, and finally report to your first submarine. Once there you'll qualify EOOW, then work on qualifying stuff up forward like OOD. Every officer on a sub is a nuke (except the Supply Officer), so at one point or another you'll likely be responsible for just about everything on the sub from the engineroom to driving the boat to the weapons and sonar.
If you have any other questions you think I can help with, let me know.