After my palm caught my fall, I have a swollen thumb, scrape at the bottom of the thumb and a tender thumb. The fall happened five hours ago. Should I see a doctor?

Must X-ray. You need an x-ray to make sure you don't have a broken bone!
Yes. First off clean the scrapes with soap and water, peroxide. Then rice- rest, ice, compression, and elevation. But you should see a doctor. There are many things that can injured from a fall like that, some of which would need special treatment. For example, there are fractures at the base of the thumb into the joint that will be a problem if left untreated. Same for certain types of wrist fractur.
Not a bad idea. It's a good idea to get injuries from falls like this checked out, especially if they continue to hurt after rest, ice and ibuprofen, or if the thumb is particularly discolored or deformed looking. The main concerns are the blood flow or damage to one of the nerves in your hand, and proper mechanical functioning of your thumb. But if you can use it normally it's probably ok.
If it doesnt improve. It's likely just a contusion or sprain, but if the pain doesn't improve by tomorrow, see your doctor for a check up. Keep the scrapes clean. Monitor for redness or signs of infection (drainage or pain).