A three year old hen with no eggs for a year? Before you start weeding out low-producers, you may want to do a parasite check. What breed is she? Even the production mixes, like the stars or comets, will continue laying past the 18 month commercial culling point. They just slow down some...
Yes to both questions. There are lots of rabbit wormers available online and at your local feed store. If you want to be specific, you can take a poo-sample to your vet for a float test. Sometimes your Ag extension service offers the service as well, but either way, it's not usually too expensive.
Definitely a Silkie mix. Going by the poofy back on the hen, I'd venture to guess there's some Cochin in there, too. Regardless of the mix (and to me, they look like mixes, not a breed,) they're VERY striking. That's a lucky find!