(1) If STA-21 is unavailable as an accession route, what are my remaining options for officership (assuming I enlist): OCS? or is this just a non-college NUPOC without the fancy bonuses?
Pretty much OCS as far as I can see. You could not be more competitive for the nuke officer position by completing any portion of the nuke pipeline, generally. Now, they are always hurting for nukes, so there is the possibility that applying for OCS and interviewing for nuke can happen (as far as I know). Generally, the interview comes first, then selection, then OCS. So, from inside the nuke pipeline without going through another officer accession program I am unsure of how to get selected for nuke before going to OCS, or if it is possible after OCS.
Theoretically, you could be frocked in an emergency, this is when your paygrade stays the same but you wear a higher paygrade or rank and are respected as such. This will probably never happen.
There are Limited Duty Officers, they are enlisted (generally E-6 or higher) who become officers. They cannot command the ship and are limited to certain duties. They apply for this position, generally for nukes, this is a radiation control officer.
You could also theoretically apply for a warrant, after becoming chief (E-7 or higher). Instead of being "appointed by the president and confirmed by the senate" and receiving a commission (yes, this is the formal process; you never hear about it though, but it happens every year) you are given a "warrant" to serve "by the president." Obviously, the president again has nothing practically to do with this, but it explains the legal reasoning behind the difference between chief-warrant officers and commissioned officers. You become something above enlisted and below and officer. You never really see these people in the navy because there are other routes available and pursued. There are not many.
(2) Assuming I am already enlisted and undergoing nuclear training--will I need the extra semester of Calc II & Phys II to even make pick-up for OCS an option?
Most definitely. This is the only way for the navy to evaluate the skills of people of disparate engineering and science (and non-science, non-technical) origin. Its my understanding that at one point in the past some people were allowed to move from enlisted training to officer training if their grades were exceptional enough without ever going to college. Perhaps I am wrong, but a reliable mathematics professor (O-4 air force, retired) had a brother who told her about it, he was in the nuke pipeline then back in the 80's. They aren't hurting like they were, no cold war now. Further, most of the people who back then would not have gone to college now do so, because its more affordable. This may provide a sociological explanation of why it would be unreasonable to think this possible today, generally.
Anyway, you could earn these quickly. You'll need good grades and enough understanding to
use it. However, if you were to enlist right now, the time spent in between going to boot camp and now could be equal to the time it takes to finish those courses, especially if you were to do a B-term on quarters (for winter) or an A-term on either quarters or semester in this summer. It would be challenging to complete these courses during the nuke enlisted pipeline, you need to do your best there.
You know, the energy that made you choose your non-technical major should be the source of energy that makes you want to be a nuke officer. There is no shame in choosing social sciences or humanities. Don't deny that it reflects who you are or who you were. That same energy can allow to change you direction, after all, its the source that led you to choose you path in the first place! It doesn't make sense to resent that decision, now does it?
I hope this helps you on your quest!