My wife and I are taking our oldest daughter to London this summer to celebrate her college graduation. We are going to be in London for 2 days at the start of our trip, we will then take a week long tour around the country, and then spend one more day in London at the end.

None of us have ever been there. I have been planning our trip and was intending to use public transit to get around but I am confused by some things I have seen. If any of you are familiar with the London transit system, can you answer these questions?

1. What is the easiest/best way to pay for transit?
From what I understand we can use an oyster card or our own tap-n-go credit card. Either way we will need three different cards, correct? Assuming there are no foreign transaction fees on our CC are there any advantages or disadvantages of either method? Or is there a better method? I do not think our phones will all have data service so I am not planning on relying on a payment app. With an oyster card do we have to guess how much we will pay before and then get a refund if we do not use it all? Is that a hassle or not a big deal?

2. Do routes run in reverse?
There is a very helpful site that helps me to plan routes (rome2rio.com). Our hotel is near Shepherd's Bush Station and it gave us this route with one transfer and a cheap rate to get there from the airport. (https://www.rome2rio.com/map/London-Heathrow-Airport-LHR/Shepherd-s-Bush)
However, when I try to get a route from Shepherd's Bush back to Heathrow the routes are all long, expensive, or complicated. (https://www.rome2rio.com/map/Shepherd-s-Bush/London-Heathrow-Airport-LHR)
What gives? Can't I just get back on the same trains but going in the opposite direction?
The official website (https://tfl.gov.uk/plan-a-journey/) is even less helpful. It seems to only give me one route that takes almost 2 hours. And sometimes it does not return any route at all.

3. What do you consider must-see?
Obviously we would spend more time in London if we could, but we essentially only have 2 and a half days there. What are must sees? We are planning on the British Museum and Tower of London. What other things would you not want to miss? And it seems that most of the tourist things close around 5 or 6. Are there good things to do in the evenings? We might catch a show, but want to see what other options there are to maximize our short time there.

4. How early do we need to get to Heathrow before our flight?
Google tells me that for an international flight we should arrive 3 hours early. Is that good advice? Usually in the US you are safe with 1.5 - 2 hours early, but I know that it can depend a lot on the airport.

5. Any other general advice?

Thanks for your help!