Yes, the parts under the saddle can easily get the 'dust' from any dust bath and if proper care is taken with the saddles there's very very little chance of getting lice (we've actually never had a customer get lice from having our saddles on). They do need to be cleaned and once the feathers...
No, a hen won't have any trouble dust bathing and getting hung up on anything would be pretty rare. The elastic goes under the wings and is nowhere that it will get caught on anything. Your hen's wings will sit on top of it :)
Sometimes they flip out just because it's something different. Try putting it on at night when they're roosting. Chances are in the morning, there won't be any racket and they'll go on like she's not wearing anything :)