I would use the AA arrivals lounge. It’s more straightforward. They also have loads more showers than any of the airside lounges (dozens of them - so no wait). The breakfast is very good (not better than in CX F lounge, which you can save for later in the day, but as good or better than BA F Lounge).

I don’t think any of the lounges will store your bags without you being present in the lounge, but you can ask. If not there is a luggage company at T3 who will store it for you for the day for £12.50 a bag. https://www.excess-baggage.com/

On the day room option. You can transfer between terminals airside. When you arrive follow Flight Connections and get a bus to T4 or T5, then clear immigration in those terminals.

Landside you can transfer easily and for free by the Heathrow Express or Connect trains. T5 will be easier as the trains go every 15 mins.