As a European, I'd favo(u)r PCR because the final D in Picard is supposed to be silent. P-KAAR. Though that ship has sailed and sunk decades ago. Memory Alpha is already going with PIC, btw.
I'm not sure. S31 ("Ess-Thirty-One") sounds odd, and I think just Section 31 would be better, but it does have the benefit of separating the show from the organization. It might grow on me and is without a doubt what everyone else will use.
Memory Alpha decided on an abbreviation convention long ago and will probably never deviate from it. Multi-word = initialism, one word = first 3 letters, If we had "Star Trek: Quark's Bar, Grill, Gaming House and Holosuite Arcade" and "Star Trek Assignments", I'm sure they would call them QBGGHAHA and ASS without missing a beat.
But it has more than one syllable. That makes it harder for some to spell, never mind enunciate. Just as long as it doesn't become "S-TP". And that depends. But likely will if the first couple of episodes fall into the terlet.