Can my Dell Inspirion's recovery drive still work after overwriting it with another OS?

As long as you are writting on your backup drive, you should have no problems. You can wipe the puppy out, put what you want on it, wipe it out again, and make it a backup all over. It is as easy as tossing the ball, catch it in the cup, drop the ball, toss it, and catch again. And if you miss, don't worry, the ball is attached to a string that is attached to the cup. Ball-n-a cup!