These are always the first things people say when I tell them I don't have a credit card. Far as I can tell they're both completely false, at least as of recent history. I rent cars and book hotel rooms on my plain old debit card all the time.
You can rent a car with a debit cards in many cases. Avis has a page [1], but it's lacking specifics, so you probably need to call the desired rental location to verify. Enterprise seems to say they accept debit cards (with some restrictions/deposit requirements).
Doesn't seem like a can't, although it might be in some locations.
Most hotels require one too.
This is for any situation where your liability may exceed your cash on hand.