As for the title of "God of war" think about it from Zeus' perspective:
in GOW1 Kratos killed Ares, and then went on to do Ghost of Sparta, and only after THAT, was Kratos named the new GOW. There was plenty of time for Zeus to look for a new GOW.
But in GOW2 Zeus tricks Kratos into giving up his godly powers into the blade of Olympus, forfeiting his GOW title, and then 2 seconds later, future Kratos pops out of a portal and kicks the crap out of Zeus, and immediately after THAT, Kratos starts mountain climbing up mnt. Olympus with all the Titans.
Zeus never got enough time to look for a new GOW after Kratos, he was still dealing with an angry Kratos & Titan war.