The mussels you buy in a shop for human consumption are usually from seawater or brackish water and die in freshwater. I wouldn't add mussels that have been out of water and sitting in a shop. They are generally close to dead when you buy them, apart from being salt or brackish water species.
Dead mussels are easy to pry the shells apart. Live healthy mussels will close their shells up quickly and keep them closed. Dead ones also stink so use the nose and have a sniff, if they smell slightly bad, they are probably dead or dying.
The risk of diseases is virtually 0 from wild caught freshwater mussels. Some marine and brackish species can carry a virus but it only affects other mussels/ clams. Freshwater snails can carry intestinal worm larvae or eggs. Freshwater shrimp can carry microsporidian infections (identified by a cream white muscle tissue in the tail). But the risk of introducing diseases is more likely from captive bred stuff (ie: fish farms) like Gary said. I have never had a disease in my tanks from wild caught fish, shrimp or shellfish.
Mussels are better for Bitterlings than clams. We had freshwater mussels about 2-3 inches long in the tanks with Bitterlings.
You feed mussels and clams with a liquid marine invert food. These are usually water with dead copepods and other fine particles in made from blended up fish and prawn. You can also use algae from the glass (get a sponge and wipe the glass down, then squeeze it out over the mussels). You can add green water (single celled algae) to the tank. You can use an eye dropper to squirt some of this on them. You can use a boiled egg yolk squeezed through a handkerchief into a small container of dechlorinated water, mixed up and use the eye dropper to add some of that solution to the mussels. They do best when fed every day but need feeding at least 3 times a week, especially in a well filtered aquarium.