If I'm ever hospitalized, should I tell the doctors that I had a DVT or pe?

Tell any doctor. Even if you are not hospitalized, you should tell any doctor who cares for you that you have had a DVT or a pe.
Yes... If you have had a DVT and/or pe, this is an important part of your medical history that you should always tell the doctors caring for you in any setting. There are sequelae that can occur after having a clot and you could hs e recurrent clot. So, do not leave out this information!
Yes. Please! absolutely. This is vital information about your medical history that the doctor needs to know. It alerts him/her to the fact that you may be at risk for developing another clot, especially since you are sick enough to need hospitalization.
Yes. Because you are at increased risk to form these again.
Yes. You certainly should. This will help guide your physicians in choice of method to prevent blood clots (blood thinners vs. Sequential compression devices). Depending on other medical and family history, a history of blood clot (depending on the circumstances) may also clue them to perform some tests to see if you have a predisposition for clots, which can be extremely important to know.