In our hospital tank we have the 4 remaining minnows and the new snails and 3 white cloud minnows. We also have Hairy and Larry (the Moss Balls) and about 4 medium sized frogbit plants and a baby sword.

I need to do a one more prazi treatment on the RR minnows before they go in the big tank, then the "hospital" tank  will have the 3 white clouds in it until we get another goldfish.

Our voracious gold fish Sebastian chomped a hole in poor Hairy, so I'm afraid the Moss balls won't be staying in the big tank.

So anyway, my inclination is to remove the plastic plants, and the real plants while the medicine is in the 10 gallon hospital tank. I'm just wondering about the snails. Do I need to move them, too? Do they need to be treated? Do they carry parasites to the fish? is this even safe for them?

I would separate the snails as well, just to be sure! There should be some information with the medicine though whether it's suitable to use with invertebrates or not.

I know that praziquantel is safe with inverts but not crustaceans ...unsure about the other ingredients

Your plants will be fine

Are you doping their food as well as the tank?

put your   food e.g bloodworms in a cupful of tankwater and add a tablet. leave for a while and feed to fish