If applicants need to stay through Wednesday afternoon, for example for interviews, do they usually check out of a nearby hotel that morning or spend the night and leave the next day
What hotel? Interview candidates stay in college. The college will let you stay an extra night (or more) before or after for free, just ask (one candidate from NZ that my son met there arrived on the Friday for a Monday-Wednesday interview slot so she had time to get over her jet lag). Since flights back to the US are likely to be on Thursday morning/early afternoon, just get an early bus to Heathrow on Thursday.
What @Twoin18 said.
Also, the Heathrow-Oxford bus is super simple & very reliable- definitely the easiest way to get back & forth (besides a car or limo ofc). If the college isn’t a central one, there are taxis right at the bus stop.
Might have to accompany just to catch Christmas at nearby Highclere castle? I assume the student is occupied pretty much the entire time?
Copy that!
Yes, assume that you are surplus to requirement for the duration 
IRL there will be periods of boredom / waiting / etc for the students- but they are getting to know each other as well- many friendships have started during the waits between interviews / waiting to find out if you have been called for an extra interview
Honestly I would advise letting your kid go alone. It will be embarrassing and a distraction to even mention that the parents are around. And there’s a lot of bonding happening amongst the kids there for interview. The whole process is designed to make them love Oxford (which is of course desperately upsetting for the two thirds of interviewees who don’t get an offer).
It really is super easy to fly into Heathrow, walk to the bus station and get the bus to Oxford. If you really have to fly with them across the Atlantic (?!?) then stay in London.
No need to accompany, but I was looking for an excuse for a vacation ?
Unless you have other things to do, there is a huge amount of dead time. My son had to arrive by 1500 on Sunday and could leave at 1200 on Wednesday. For four 30 minute interviews. On Wednesday, went straight from Worcester to the Heathrow bus at Gloucester Green and caught a 1600-ish flight home. I had friends/family that I could spend time with and really only saw him for a meal on Tuesday.