If I kill someone without having a quest telling me to do so and get that quest later- is it then impossible to end that quest or does the game understand that I killed him earlier?

Yes, the game understands that and gives you credit for that. At least, in every quest ive encountered. Was there one in partcular you had in mind?

Thanks
No particular one, I just cleared out some dungeon and saw on a http://www.uesp.net/wiki/Morrowind:Mawia that it relates to a temple quest later, so I wondered how the game handles this.