As a starting point, this is copied from a separate thread but addresses several questions that I suspect are common.
I'll attempt to answer all of your questions in order:
College Prestige? -- Yes it matters. There is a tiered system of schools which is used for considering applications, as well as some leeway for discretion by screeners --- ie. even though Penn State and MIT are both the top tier, clearly if all other things are equal MIT will win out...
Intended Major? -- It's important. Having said that the requirement is that you've taken 2 semesters of Calculus and Physics (each - 4 total). If you've got great ACT/SAT scores, a high GPA, good grades in your technical courses but are a French major --- you're still eligible. Personally I was an Economics major in undergrad at the Naval Academy but was a Submariner, and know of quite a few similar cases.
Class choices? -- Most importantly get good grades in whatever you choose. If all of your classes sound comparable to underwater basket weaving or Argentinian Womens studies someone might look askance, but as long ask you're taking real coursework and doing well you'll be fine. Nuclear stuff maybe gets a slight up-check, but really the GPA is more important.
Extracurriculars -- We're looking for LEADERS and Engineers. The "whole person concept" applies, and I've seen several people be accepted despite borderline interviews based on extracurriculars. Short answer: yes, it helps. On the other hand if you can't do real math it won't help enough. It's an "on the margins" type of help.
Times to apply -- For operational (Submarines/Surface) and Instructor (Prototype and Power School) you can go to interview 30 months prior to graduating --- basically December or January of your sophomore year. You could start an application as early as the summer after your freshman year.
How long does Application take? --- It varies, but basically it depends on how quickly you complete paperwork associated with the application and associated security clearance, and if there are any issues getting medical clearance. It can take under a month. The average is probably 6-10 weeks. If there's a big issue with Medical, I've seen year-long applications who eventually got accepted.
Physical fitness -- You have to be within fitness standards to go to MEPs (basically you can't get medical clearance if you're really fat). Otherwise the only way it comes into play is that being fit will make you look more put together when you're at your interviews and with the Admiral. Right or wrong (I think right), physical appearance reflects the standard one sets for oneself and if you are obese that doesn't inspire confidence.
ASVAB is ignored for NUPOC. We look at the caliber of the university, any masters/PhD work (though this is irrelevant if your undergrad GPA is bad), your SAT/ACT scores, and your GPA -- especially your grades in math/engineering/physical science courses.
Monthly Pay/Signing bonus. Bonus is $15,000 as a one-time payment after getting accepted with a smaller bonus once you pass the training pipeline (I think this is $2500 or $3000 but I honestly don't recall; not huge but not Zero). You won't get another bonus in the Nuclear Navy until after your 5 year commitment if you decide to stay in. The monthly pay as a NUPOC will show up via direct deposit bimonthly. There are 3 basic components: Base pay (this is your taxable pay, the rest is -- awesomely -- tax exempt), Basic Allowance for Subsistence (BAS) and Basic Allowance for Housing (BAH). BAS and Base Pay are standard (you'll be paid as an "E6" initially with an opportunity to promote to "E7" while in college --- search Google for "Military Pay Chart" to see this. BAS is a few hundred-ish. BAH varies based on where you live -- in San Francisco its about $4,000/month. In Tupelo Mississippi it's surely much less. The average for an E6 is somewhere between 1000-2000/month but you should look this up based on where your school is. (all is publicly available).
Tuition Assistance --- As discussed above. Short answer, no. You're getting paid to go to school. Pay for school with your pay. ***** One caveat:
Post 9-11 GI Bill --- Your time in NUPOC will count towards retirement and towards your Post-9/11 GI bill benefits. This can be a huge benefit. For instance, Of the top 10 MBA programs only 1 (University of Pennsylvania) is >$15k/year after GI Bill benefits and the Yellow Ribbon Program (vice ~70k/year without). Two (University of Virginia and Dartmouth) are completely free. You also get a housing and living stipend while in school ---- Basically when/if you decide to leave the Navy you can take these benefits and attend the best school you get into for nothing or next to nothing if prefer that route to entering the workforce immediately. (I used MBA as an example because that's the route I'm taking; same would apply for MEM, JD, other Masters etc)
Apply > once? Depends on what you mean. You can apply multiple times, but with very rare exceptions you only get one shot to go to DC for your interviews. Occasionally the Admiral will offer for a student to come back after a semester to re-interview, in which case they can obviously return --- but this is the exception and not the rule.
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I hope this helps you and any others with similar questions.