Could the radiation therapy for my prostate cancer be giving me radiation prostatitis?

Possibly. I'm not sure exactly what you are referring to by radiation prostatitis but if you are referring to increased urinary urgency and frequency then yes this is a possible acute side effect of radiation treatment to the prostate. This can be managed easily with some medications that can be given to you by your radiation oncologist.
Yes. Radiation can irritate the linings of the bladder and urethra which runs through the prostate. It can be treated easily with medication as any other prostatitis and advice of certain foods to avoid. A urine test should exclude an infection.
Or 'proctitis'? Medical words are confusing! the "itis" suffix means inflammation or irritation. The prostate itself can swell and irritate from radiotherapy, and there are medications to treat. The rectum is adjacent to the prostate, and when irritated, or inflamed, it is called proctitis. Sounds close, is close, very different treatment!
Yes. In fact, i explain to my patients that the irritation of the prostate after seed implants comes from two components. Mechanical prostatitis in the short term because of the physical procedure of putting the needles and seeds in the prostate, and radiation prostatitis because of the radioactivity of the seeds. In general all of this improves over about three months.