Okay so I imprisoned wuunferththe courtmage, but when I go to talk to him he tells me to go to the stone quarter and watch for the real killer, but as I make it halfway through the keep of windhelm, the quest suddenly updates and says chase the killer!! And when I go to kill calixto, he is invincible and can't be killed, even in hjerim. What do I do?? I have no previous saves and I have tried over and over again with the same results

Unfortunatly there is probably nothing that you can do, in Skyrim there are glitches in just about every quest that happen to some people and dont happen to others you will probably never be able to complete the quest. i Have things that are on my quest list that are impossible to complete because they have glitched and there is literally nothing that i can do.

What is your game version? This issue was fixed in the 1.4 patch.

I have Xbox 360, I haven't tried in a while so maybe it's fixed, I just haven't tried for fear of it not

I'm on the PC and I had this option as well. Only thing to do is to reload a previous save and use a guide for this quest, seeing as it is very buggy!