If I destroy my computer, will the programs on it remain if I replace it somewhere else?
It will lose its programs. A CC computer considers itself "new" when it's placed and will have a new computer ID. (CC stores filesystem contents by computer ID, so having its ID change makes it "lose" its files, even though they're still in your world's save folder if you go looking in Windows Explorer or equivalent.)
However, there's an easy fix. Any computer (including turtles) that has a label keep their computer ID and therefore remember their contents. You can set the label with the command:
Computers forgetting their programs had been giving me a lot of headaches until I learned about labels, since I was using a turtle and a custom tunnel program. Now I have two turtles, Cricket and Grasshopper, and they keep their custom programs when I break and move them. I set their labels with:
No more carrying around floppies and disk drives in the mines!
Nothing official, just a couple of facts found online: