(1) The Vatican made no territorial claims in northern Germany at all.
(2) The Emperor's case against the various statelets of northern Germany was that their change of religion did not entitle them to absolve themselves of their sworn allegiance to him, nor did it entitle them to the confiscation and destruction of Church property.
(3) The Emperor made no territorial claims on France and France asserted no claims in Germany. France's sole purpose was to aid the Germans or the Bohemians or anyone else who wanted to break up the Habsburg territories, thereby reducing Habsburg power.
(4) Gustavus Adolphus' sole goal was to wage an inexpensive war (i.e. the French agreed to help bankroll his efforts) that would enable him to seize as much new territory as possible for his own profit.
Gustavus also had a taste for war and enjoyed it for its own sake.
It is hard to say whether he had any serious spiritual attachment to the Lutheran system or not. The rightful heir to the Swedish throne was Sigismund III, a Catholic. His sole claim to precedence over Sigismund was his Lutheranism - so it isn't clear whether his Lutheranism was a matter of intense personal devotion or political expediency.
Most of his reputation as a devout Lutheran rests not on the bloody catalogue of his recorded deeds but on hagiographies written about him after his death by courtiers and clergymen who owed their status and wealth to his dynasty.
Had religion really been his motivation for entry into Germany, it seems unlikely that he would have waited eleven years to get involved. During that time he engaged in a strategically unnecessary but very profitable war with the Livonians. There was no easy money in the German war - until the Catholic French paid him to intervene.
Up to that point, he was content to watch the plight of his fellow Lutherans quite disinterestedly from a distance.
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).