I wish that I could correct you with respect to the NC arrivals question (as this is how I generally get to LHR!), but alas, you are correct you do NOT have access to the arrivals lounge. However, its probably worth giving it a try on the outbound, as you may have a nice GA who decides to print you an invitation, and on the inbound, the worst possible thing they can say is NO.

A more LHR savvy traveler will have to answer your other q's, but IIRC last time I connected to BMI as *G, there were no showers in the lounge.....I do hope I'm wrong about this for your sake, altho I expect that "paid" showers are still avail throughout...

Best wishes for a great trip! Dave