I believe you have to wait 30 days between apps. I would wait at least until you pay your second bill at least. And yes, it is true that for existing cardmembers, a denial is always just a SP. If you get approved, they do a HP just to double check. Just make sure you apply while logged in to your account!
It's 30 days if you're denied (application will be automatically cancelled); however, for approvals there's no such requirement. I'd probably wait the extra month anyway as scientifics suggests; however, there was another thread yesterday which discussed the spend requirements to really make the PRG worthwhile; I probably need to go napkin math it out at some point as I was frankly shocked by the findings but I'd suggest taking a look at it.
Edit: for reference and so I don't lose it (memory failing these days), all credit to Cdnewmanpac and CreditScholar: here
Thanks. Yea, I think after doing the math in my annual spending habits, the PRG is just not worth it. I am not going to spend at least 30K in a year to make he PRG worthwhile. Thanks for the link Revelate!