The vet should give you guidelines on how to get the stool sample. I believe it should be okay overnight in the fridge.
I use Greenie's pill packs when I need to give Thor medicine. Basically they are very soft, flavorful treats with a hole poked in the side where you hide the pill. They are very malleable, so I only use half a treat, and mold it around the pill. Thor always swallows it in one bite. He gets very excited when he sees that bag come out, he has no idea I'm slipping him drugs.
Oh, and for the slime (poor Uni!): can you get some litter and have her go on that? Then you can just scoop some litter along with the slime.