Connecting gets your address into the ruling, and nothing else. There is no claimant bounty in these contracts and none is implied: a reward would make beneficiary a value whoever pays the gas could redirect. That is a real weakness of unrewarded fraud proofs — nobody is paid to go looking — and it is a deliberate one.
checking which evidence has already been ruled on…
You cannot ask for both. Recording consumes the claim id, so asking to slash afterwards would return AlreadyClaimed instead of a real answer.
Submitting a claim needs a wallet, because it sends a transaction. Reading needs nothing — every record on this site, and the whole of the landing page, is readable with nothing installed.