skybiggles
The answers to your questions are airline specific because of different scheduling arrangements, and even in the same airline there are different arrangements for Cabin Crew and Flight Crew.
Generally the pilots with UK operators tend to get two "local" nights off on arriving on the US West coast. They arrive PM US time and depart PM 48 hours later, arriving in London around mid-day the following day ( e.g. depart UK Monday, arrive back in UK Thursday) - so yes there is time to fit in sightseeing or whatever else turns you on - or sleep.
Time off in the UK after depends on the way you are rostered but it can, legally, be just one night off in the UK before flying again.
It's not unusual to do several West Coast round trips a month - how many depends on the airline's rostering.
You generally carry a relief pilot.
The flights can be tiring, but with the extra pilot you at least get a break ( perhaps 3 hours), which takes the edge off the fatigue a bit.