Disclaimer: I am not now, and have never been in the Navy. I spent 18 years as an Air Force flight medic and Army Nurse. But I have spent two tours deployed with the Navy/FMF, so I am somewhat familiar with the issues involved here.
ACLS, PALS, and some other merit badge courses are available on many military bases, but it is certainly not universal. It depends on the local unit, whether they have any instructors, and whether they have an aggressive educational team. I did my last couple of PHTLS certifications at Fort Sam Houston. I've done ACLS on several different Army and Air Force bases. They are also available online, and many, many deployed personnel are taking them that way. So are EMT and EMTP recertification courses. You shouldn't have any problem keeping up with those (even if you expire, you can just refresh and retest without repeating the course). But even in a classroom, I wouldn't count on much education out of them. It's rare to find an instructor teaching a card course -- or even the students taking them -- who are interested in anything more than getting it over with.
As for taking paramedic school while in the military, I can't think of any reason why the Navy would object. However, the uncertainties of military life may well make it difficult to complete the program. Deployments, rotating shifts, alerts, and other such things can cause you to have to drop out, which can be a real bummer. I have, however, taught active duty personnel who were indeed able to complete the program.
Good on you for going for the degree. While the piece of paper itself does not necessarily make you a competent medic, or any better than other medics, the education you receive to get that piece of paper does. A medic without those support courses is less than half the medic he could be with those courses. What will be a lot easier to complete while on active duty are those support courses. What is required for the degree varies greatly from school to school. Some schools have a lot of relevant prerequisites. But many others just pad their degree program with pointless nonsense. Obviously, if you know where you intend to complete your degree, then you should focus on their specific list. But regardless of where you go, there are certain courses that -- required or not -- are essential for a complete foundation for medical practice. Those include:
Anatomy & Physiology 1 & 2
Microbiology
Chemistry
Intro Psychology
Developmental Psychology
Intro Sociology
Communications
Algebra
Of course, you're going to end up needing English, History, and other non-science support courses to graduate. And they are equally important to your professional development. However, those are courses that are not immediately necessary for building your foundation for the medical courses. And they can easily be taken later on, even during paramedic courses, without being too much of a distraction from the medic courses.
You might, however, check into online classes for History and English, Sociology and Psychology, and even Algebra, if independent study works well for you. There is the potential to receive a quality education in the non-sciences through distance learning. And that would be a few less courses you'd have to worry about on down the line. And you wouldn't have to worry about dropping them if you get a PCS, deployment, or schedule change.
As for flight medic, I've never seen anyone requiring a BS in Health Science for a flight medic. The flight medic jobs seem to go to those with major street experience, serious educational credentials, and those who "know somebody". But it's pretty damn hard to find a street medic with a BS degree in anything, much less Health Sciences. If you follow the course recommendations I listed above, you're going to be qualified for almost any Nursing school, so that gives you an option that is much, much more likely to get you into the air than a degree that doesn't carry a license with it. There are some excellent BS degree programmes for paramedics, but unfortunately, there are others that are very poorly structured, resulting in little to no medical education beyond your initial paramedic education. If I'm hiring flight medics, your Health Services degree, which typically has a lot of non-relevant courses, is not going to be half as impressive as a BSN or Respiratory Care degree. And RTs who want to fly are in a lot more demand than nurses who want to fly. Combine that with your AAS in paramedicine, and you'd be at the very top of any hiring list. And hospital experience is really the only way to get the critical care education and experience you need to become a true critical care medic. Hopefully, you can get some of that while still in the Navy, depending upon your assignment. But working the base ambulance isn't going to do it, that's for sure.
That said, flight medic isn't as glamorous and sexy as it is made out to be by all the admiring rookies. That helicopter is just another ambulance. And the people on it frequently aren't any more qualified than the people on the street. They are frequently overused for too much BS by incompetent EMTs and medics who don't know how to properly triage or care for their patients. Consequently, a very large portion of your patients didn't even need your services, just like on the streets. A lot of flight nurses and medics die a horrible death every year flying total BS runs that should have never been dispatched in the first place. I've lost a few friends that way. There's nothing heroic about having your body ripped apart and incinerated over someone's whiplash wreck, leaving your family without you. It's at least as great a job as being a street medic, for sure. But I'm just saying to do some serious research into it before you get crazy pursuing that goal. All the sexiness wears off pretty quickly.
Best of luck, Greg! You've got a good while to go, so take your time and use it wisely. Map out a solid plan to establish the best educational foundation you can before entering medic school. Then never stop being a student. If I can help you with anything, shoot me a PM.
Alek