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.
If he had the Soul of Ares (non-canon extra from GoW1), then that wouldn't be out of the question. Entries after it could have done so while giving us his transforming weapon (Axe, Hammer and Sword) and spells (fire based).
>still the god of war
Kratos became as such in name only for GoS. After it he became a full god (in all rights) for GoWII (then became a fallen god).
Lord: >a God
You didn't mean to use a proper noun, so why is this capitalized?
>surpassed the gods
Nyx would stomp him into the ground.
>couple of titles
Captain Kratos, General Kratos, Marked Warrior, Cursed Warrior, Ghost of Spartan, Fallen God, Cursed Mortal, Slave of the Gods, Champion of the Gods, Fist of Ares, Champion of Ares, Champion of Athena, Godslayer/Slayer of Gods, Death (and all that jazz), God of War, Lord Kratos, etc. It is a very long list.