Depends on your show and if they're governed by USEF's rules. But some schooling shows who are affilliated with USEF still have their own rules for certain classes so I'd call someone in the know: the show steward, someone from the barn hosting the show, an officer of the horse show association, etc.
1. At USEF governed shows those showing in pony specific classes must be juniors, 18 and younger. For the most part anyone can ride a pony in any other class. However, I've shown at schooling shows (governed by USEF affiliated organizations) where professionals can't show ponies in open divisions (which basically means pros can't show ponies at all but amateur adults can show them in open divisions).
2. I just looked at USEF's rule book for jumpers and it says this:
-classes offering less then $1000 - no martingale restrictions
-classes $1000-4999 only standing or running martingales used conventionally
-classes $5000 or more (and all classes restricted to young horses)- only running martingales
BUT, I'm almost positive I've read a prize list somewhere that said that at that show anything above a certain level (not money) no standing martingales are allowed. And I think it was like, Level 3s or 4s or something (i think anything above 3"9 ish). But I can't think of a single jumper I know, even lower level, who uses a standing martingale.