Those are great questions, some of which are a mystery!
First, yes, absolutely, you bet, PIs are very difficult to get and are very competitive. As far as the process, it depends on the PI. Animal Programs are handled very differently from, say, management PIs. I applied for some AP/PIs and mine was all handled through email. It stated that if they needed more information, they'd call, otherwise I'd know via email by May 1st.
Last semester, for the management internship, there was a screening interview, then a first interview where my application was then forwarded to the specific departments, then I had a second interview (all over the phone) with the merchandise management folks. I had not done a CP, but from the way things went down, I'm pretty sure I would have had an offer had the economy not tanked and the internships cancelled.
Communication was non-existent after the 2nd interview until the sad email about the positions being cancelled. From what I saw, they called with an offer, emailed a rejection.
I've applied for 2 non-Animal Program PIs this semester and have heard nada, nothing, zip. Not even an initial interview, which surprises/confuses me. I'm currently on a CP, so I don't know if I'm not being considered at all, or if they already have enough info from last semester's interviews to move things along. I may be calling the PI recruitment office this week, just to see what's up.