Current VFA LSO here so I'll give my $.02. On my cruise each squadron ideally had 4 LSOs (this varied as senior guys left and they trained up new guys, but 4 was the goal). That meant we would wave every fourth day which also meant we didn't stand SDO. We did stand Romeo still and being an LSO really had no bearing on ground jobs, other than that the senior one was the Head LSO (not taxing and wouldn't be considered a primary ground job). I would say being a paddles is generally sought after because it does open up interesting doors, it's fun, and getting out of SDO on the boat is a nice perk. Selection is a mix of timing, performance, and personality. The timing has to fit for the squadron, you can't be a liability behind the boat, and you have to be able to professionally debrief your peers and your seniors (up to O-7 in some CSGs). Basically, pass the "good dude" test.
More specifically for your questions:
-Most guys/gals will typically "rush" by spending time on the platform during flight ops and show an interest. Squadron timing may dictate taking someone who doesn't seem interested or not taking someone who really seems to want it. I've not heard of anyone turning it down.
-Since there are 4 LSOs per squadron, there's typically not an issue filling in. Each squadron has a representative on the platform every fly day, so there are typically 2-3 LSOs in the peanut gallery that don't have specific job on a given recovery, so being limited on bodies isn't usually an issue. Jobs/roles usually change every recovery to get people experience/OJT
-There's a formal ground school that's required for squadron/wing quals, but it's mostly OJT. You can easily have the skills and experience level of a wing qual without the ground school, but you'd still need the school for the qual.
