Can I book a roundtrip flight with a visa that expires before the return flight? (I understand that I will have to extend the visa while there.) The visa has too much time on it to review, plus the US Embassy here is not really taking new applications right now due to COVID-19. I would like to return a month after my current visa expires. Will I be allowed entry and allowed to fly?
You can book a flight without having any kind of visa.
Airlines do not check that you have a valid visa when you book a flight, because lots of people choose to buy a flight before they get the visa.
They will check your visa when you board, and if you don't have a visa by then you will be denied boarding and not given a refund.
Whether you are going to get a US visa, or be permitted to enter the US with or without a visa will depend hugely on the state of Covid-19 at the time.