Anyone done the LHR-LGW shuffle recently?

I'm arriving on an AA flight early morning to LHR, leaving three and a half hours later on an AA/BA codeshare out of LGW. I was wondering:

1) will AA check me all the way through and give me a boarding pass for the BA codeshare flight?
2) I'm not planning on checking my luggage - will I have problems at LGW when BA sees the rollaboard I always use when flying on AA? Do they have restrictions on carry-on weight for in-europe flights?
3) what's the best way to get from LHR to LGW? I'm looking for little to no hassle, though I'd rather not spend an arm and a leg....
4) Is there an arrivals lounge at LHR or a departures lounge at LGW that I can get into as EXP to refresh myself (presuming the transit between airports doesn't take the full 3.5 hours)?

Thanks in advance!