I have the Kindle app on two computers and both computers show the same purchased books. I want to delete the app and its content on my oldest computer. Will the books still appear on the new computer?
Yes, the books will still appear within the Kindle app on the new computer, provided that you are signed in to your account.
I believe you can be signed in to the kindle app on up to 6 devices, removing one device will not affect the books you have purchased (you may just need to re-download them on to the new device).