Did daedric princes exist in Arena? And if so, did they really play any important parts? I have played TESIII and IV only.

No, the daedra were introduced in Daggerfall.

Some of them where present though the term daedra wasn't invented yet. For example Boethia is referred to as "the dark elf goddess Boethia" in the artifact quest for the ebony mail.

thanks...one of the best ES inventions ever, IMO.