I said they could hold the trial if they wanted, so I'll leave out the first bit (no precedent for convicting was set, because that person wasn't convicted, so there was no standing for further adjudication).
The senate can pass a "resolution" for anything it feels like. Resolutions aren't legally binding. The government could technically pass a law that says "Fuck Trump, he can't run for office" but it would be extremely unconstitutional. Right off the bat it'd be a total 14th amendment violation, second off it would violate the qualifications clause for who can run for office.
No un-packed SC is going to go for that, especially the current lineup. They've already shown they aren't going to be puppets for Republicans.
Congress with the power to decide who is and isn't eligible to run for office on a whim isn't something the SC is going to be too keen to endorce, nor would Republicans support it because it would be immediately used against them to bar all right wing candidates from office once the Dems nix the fillibuster.