You can't request time out and you'll be working somewhere suitable if you're downgraded medically but don't expect that to be long term. It all depends on the reason for the downgrade and projected timescale to reach full fitness again.
There are no permanent roles for those unfit for sea, you're just taking up a billet which someone whose due shoretime can't use. Long term unfit means a medical board to assess you and possible discharge medical.
The only permanent inboard people I know of, who are unfit for sea, are FTRS personnel.
Untrue. I know a few regular ratings (mostly SRs) who are permanently non deployable following med boards and just take up gash second line roles or instructor roles.
I don't agree with it, but with the manpower shortages in certain branches it does happen.