so, god of this world and jason, would have survive a battle be you can use their abilities, correct.
This is a newer concept, so I expect to see a lot more confusion about how TC characters work for awhile. What you are describing is what happens with 'normal' characters that are not Territory Class. However, any character that is TC activates each time they hit battle
and each time they hit a territory (what browarod posted is the definition for those characters).
So, if you place The god of This World into your territory, he actually activates then, and stays active until he leaves territory. It is one of the advantages of these new characters