There are some that don't, for example
pomóc - to help, móc - to be able, biec - to run, piec - to bake
Here it is Dative. but it can be locative. The hints are crazy. mojej also has hint for Genitive and Locative but not for Dative.
Yes, but better to think of it as primarily "have to" as "nie musieć" means "don't have to". And in any case "have to" is used more generally than "must".
I never though about how there is no "do not must." It is neat what you learn about your own language while learning another's.