What was Dumbledore thinking?

He didn't tell him to kill him because of the wand. He was already dying because of the curse from Tom's ring. Snape contained it to his hand but it wouldn't last there, and would spread. Both Snape and Dumbledore knew this. Knowing that he would die anyway, he wanted to save Malfoy from cementing his fate on the wrong side by killing Dumbledore. Even though he didn't want to do it, Dumbledore knew that Malfoy wouldn't be able to come back from that. Snape had to be the one to do it, if nothing more than to get Voldemort to trust him completely. He didn't die to protect the wand, he died to protect Draco and to give Snape the foothold that he needed.

Dumbledore had made peace with his death, and he knew that Snape had done so too. Snape's acceptance of the unbreakable oath to help Draco meant Snape was doomed either way. They were just doing what they could with the rest of their remaining days.

J K Rowling could probably confirm that Dumbledore had already foreseen how events would play out up to the final duel.

Snape was on Voldemort's side (supposedly, in V's eyes).