If you (Nico, that is) are already in DEP with a Nuclear Field Program guarantee, there is no further testing necessary. The verification of your eligibility has been done by the classifier ("job counselor") at MEPS and if there was any question of your qualifications, you would not have been a Nuclear Field guarantee. So, if you read and fully understood the contract you signed, your recruiter is not likely to know much more about the specific requirements.
I spent all my time in the nuclear navy on submarines and was offered the opportunity to volunteer when my DEP contract was written. (Actually, it was more like, "Do you want to volunteer for submarines?" and I, not knowing much about the Navy, thinking that it was the normal thing to do, replied, "Sure, why not?") I'm 5' 0" (how do I know? because I didn't need a height waiver, so by definition I must be at least 5' tall) and my early experience was that most MMs and ELTs tended to be average height or shorter while most ETs seemed to be big and tall.
I would imagine that you could volunteer for submarines at any time. You could probably do it now, if you really wanted to, but it probably would involve writing a new contract and you'd have to go back to MEPS to do that. I say that, not from any direct knowledge, but because, 20+ years ago, submarine volunteers would be treated a little bit differently. Typically, the submarine volunteer would have his wisdom teeth removed. I don't know if they do that now. Yes, you could do it after A school. You'll probably make an appointment with someone called the "career counselor" and he or she will help you with that. In any case, if you are going to do it, I would think you should complete the paperwork prior to going to prototype, because the detailers (the people who assign individuals to their ultimate duty station) probably have projections of how many sub and surface sailors they have, and if they don't know you're a sub volunteer, depending on the needs of the Navy, they might already have assigned a job for you on a surface ship.