Undeniably, you seem to have a rather clear picture of what happened.
However, I wish to correct your eyesight just a little.
To begin with, The Vatican itself didn't make territorial claims to Northern Germany, but the Habsburgs (Spain, Austria and allies like Milan) did and received support by the Vatican.
The “Holy” Roman Emperor never was a religious nor a political institution to which Northern German states owed any obligation whatsoever. Perhaps in the sphere of Wagner operas they later on did though.
The Habsburgs made no claims to French soil. But it was apparent that the Habsburgs were trying to encircle France. Remember the Netherlands was Spanish territory at that time.
Gustavus Adolphus was indeed a true warrior. From his own perspective, God gave him the throne of Sweden as well as handing him opportunities of expanding his kingdom eastwards as well as in a southern direction. What would have been a better approach than waging the wars he did? Letting the Habsburgs or perhaps the Russians gain total control of Continental Europe?
Thank you for the contribution you've made to this thread, but rest assured I'll continue remembering everything that is right about Sweden and Swedish history, especially after having eaten a Gustavus Adolphus pastry on the 6th of November (but honestly, I doubt I'll ever eat again - I've never had anything as rich in calories as the cakes I made today).