At its height, Sweden held land in current Germany. Those lands were in Pomerania, which is neither a very prosperous region, nor directly connected to Sweden. How did Sweden ended up controlling Pomerania? Why Pomerania and not lands in Denmark, or Norway for instance, or more prosperous towns like Lübeck, Kiel or Hamburg?
I'm on my phone right now so apologies if my comment doesn't answer all your questions.
The Swedish involvement in the region of Pomerania basically began during the Siege of Stralsund in 1628, when the city of Stralsund got allied with Gustavus Adolphus of Sweden (they also got allied with Denmark) to resist occupation by the Holy Roman Empire. This also marked the entry of Sweden into the Thirty Year's War.
In 1630 the Treaty of Stettin granted military control over Pomerania to Sweden. That's how they ended up occupying Pomerania.
To answer the question of 'Why Pomerania?' - while it may not seem very prosperous from today's point of view, Stralsund at the time was a Hanseatic city with an easy to control and defend port in comparatively little distance to Sweden itself. This meant that it was a good source of raw materials, a well frequented Hanseatic port and could ultimately be used as a bridgehead to launch attacks further south into Germany. Controlling the surrounding region (Pomerania) means you also control it's resources, so the region had a huge tactical value for Sweden.