Transportation in and around London for 7 days

Do some price matching on cabs vs underground passes, but I do believe there's a 3 day unlimited travel pass for underground/busses.

When traveling outside of London, I booked individual train tickets. I think they were as low as $15 each to go ~2 hours away.

You will want to travel within London by the tube or bus or by walking. The fastest is the tube. The most scenic is the bus because you are up higher than in a cab, and can look at landmarks and parks as you go by. Pick up a tube map and also a bus route map when you get into London, available at tube or train stations, and at the Tourist Information centers. I will leave the rest of your questions about rail for our rail expert to answer.

Your cab idea is a possibility, especially if you check prices and they compare favorably to all 4 of you traveling on an Oyster Card on the tube. I cannot answer that for you. However, the great thing about the real London black cabs is that their drivers all have to pass the test called "The Knowledge" and know all the history and landmarks of London. They will talk the whole time you are in the cab, telling you what you are passing and its history. They are a great source of information about London restaurants and pubs, too; so ask where they recommend eating and drinking and take notes if you go for some cab rides!

I applaud you for doing so much research for your trip! You will also find it helpful to begin reading previous threads here on this website about England. Go back to the "England" page, and keep clicking on older threads and reading. Or enter "Cotswolds" or "Salisbury" or "Hever"or "Stonehenge" in the search box above to pull up previous threads that may help you in your trip planning.

Of your ideas for days out, I 'd rate them ...
1 Oxford and Blenheim Palace (Warwick Castle was so so for me)
2 Salisbury Cathedral and Stonehenge (not been to Highclere - may need car)
3 Leeds Castle (Hever is good too but difficult to get to by public transport)

Thank you so much for your help. Believe it or not, I didn't realize there was a search bar for topics so that will save everyone from having me repeat questions that were previously discussed. Good to know. :)

Also, sorry if this is too obvious but it would be an expensive mistake: Leeds Castle is in Kent and in a completely different direction from London to the city of Leeds.

Yes, I caught that the other day as I researched . Thanks for the heads up.