Yeah places are hiring, but what do you actually have to do to have them look at your application?
When I was at the end of P2, I called one hospital like 3 times. Apparently their hiring process went through HR. She kept saying she'd look at my app but she never did. End of P2 was too late? Why can't it just be simple, where you send your resume/coverletter, and they then call the most qualified applicants for interviews a few weeks later? Why do you have to call and bother people several times a week for a month to get noticed (only to be given a job offer 6 months later when about to go on rotations, so basically too late?
Note, my response is going to be institution specific, YMMV elsewhere. End of P2 is pretty late, realistically it takes about 2 months to get from hired to on the floor working (drug tests, BG check, child/elder abuse clearances, orientation, extensive training, etc...) So the latest we will take is 2nd semester P2 since that effectively gives us one full summer and all of P3 before said student goes on rotation.
If we hire at start of P3, you'll get 6-8 months of usable (but non-FT) time before said student goes on rotation (and they change to weekend only availability). That's tough, you need them during the week (during their vacations/etc...) to train. At that point, it's not worth it, you're investing hours and hours training and only getting 6 months out of a student? If I were hiring, I'd rather hire a P1/P2 and get >2x usable/non-rotation time for the same amount of training.
In terms of applying, calling some place you applied to 3x is a bit overkill. Sometimes they'll solicit interns before a budget is in place to pay for them, it's really a timing issue vs. whether you're qualified or not (you're an intern, unless you drool during the interview and/or insult someone, everyone's qualified just the same). I did one follow-up email 3 weeks after submitting and that was that. DOP's/HR people are busy.
As for your other questions, welcome to the real world, this is how hiring works at every place I've either been hired/not called back from or did the hiring for. We don't give a crap about you, you are disposable and not a part of our company, therefore we aren't going to conform to your expectations on when we should call you back, etc... we'll send rejection emails if we feel like it.
Also, those positions might not even exist in the first place, those doing the hiring might already have candidates in mind that applied "behind the counter" by in-house referral or other query. Our experiential director would send us emails saying this-or-that hospital is hiring and to email so-and-so -- I'd go to check online and there would be a listing there, but was told to just email directly.
Hope that gives you insight.