I have used several of the save slots during the game, but they keep disappearing the next time I play. Now only my oldest save is there. I create a new one for each mission, so now I'll have to replay from the second mission.

Anybody know why this happens, and if I can do something to prevent it from happening?

That happened for me as well, hours lost playing this game. I can't find any preferences related to theme hospital on my Mac either... any pointers?

Yup, it's because the nickname is essentially an old fashioned way of doing separate player profiles. Instead of keeping a list of profiles it just expects you to remember whatever name you decided to use.