I will be flying from DFW to LHR on a redeye leaving ~4PM Dallas time and arriving ~7AM London time in F (Flagship First) on an AA flight to T3. About 6 hours later on the same reservation/itinerary I will be flying out on a codeshare flight operated by BA to NAP in the main cabin (not club). The arriving flight from DFW comes into T3 and the departing flight leaves from T5 according to the BA website.

Questions I have -
Will I need to get my checked bags from the flight going into T3 and go back through security at T5 since I am changing terminals after arriving from the US en route to Italy? I am hoping not so I can actually enjoy the T3 lounges first before hopping to T5. The next question directly relates to this.
As I arrive in T3 on a TATL flight in F would I have access to all of the partner lounges in T3 (AA, CX, BA, Qantas)?
As I depart from T5 after arriving on a TATL flight on AA in F, if I am not mistaken, I would not have access to the Concorde lounge in T5. Correct me if I am wrong on this.
Assuming no access to the Concorde lounge and given I am arriving in T3, looking for thoughts on the BA Galleries First lounge in T5 versus the 4 partner lounges in T3. Will be having both breakfast and lunch between all of these lounges. Would also ideally shower in one of them. Is the CX shower the most luxurious generally? Anything that the BA F lounge in T5 offers that sets it apart from the 4 lounges in T3 I'll have access to, whether showers, food, drink, peace/quiet, or anything else?

Hopefully I can get a good game plan together for an enjoyable few hours in the airport. Traveling with Mrs. Rossodio for leisure and looking to relax...