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1a) You will show your passport at airline checkin, so they know you (probably) have the right to enter the UK. As a US citizen, you don't need a visa.
1b) You will not go through US customs or CBP when leaving the US.
1c) You will show your passport to the UK border control in LHR.
2) If you're ticketed on United the whole time, your bags will almost certainly be "checked through". You'll drop them off in PIT and pick them up at LHR.
3) 50 minutes to deplane, go through immigration, pick up your bags, go through customs, and catch a bus in LHR is probably not enough time. Fortunately, there are a wide variety of ways to leave LHR and go wherever you want to go.
Couple more tips for international plane travel: