In the third год Harry almost got the Dementer Kiss in the book...

I think both. This is because, the dementors suck one's soul out. Voldemort's soul is mixed with Harry soul. If only Voldemort killed Harry, His soul woul die. this is all because of lilly's sacrifice. Harry can't die while Voldemort lives. Since this only happened in the third book and Voldemort hadn't taken in Harry's blood and Lily's sacrifice, part Voldemort's soul mixed with Harry's, would've vanished is the Dementors kissed him.

That is very very good question. If Ты know anybody who knows how to contact JK Rowling, I would contact her and see what she has to say.

Personally I think the Dementor would take both souls, but they also might just takes Harry's and leave the piece of Voldemort's soul, since Harry was the one they were after.