Most maps show individual German kingdoms rather than the Holy Roman Empire. Did regions within the empire have their own armies? If so, did they ever declare war on one another?
Yes they did, very often.
The Holy Roman Empire was very decentralized and weak as its own entity, the Emperor was usually just the head of state for one of the smaller nations within it voted by important officials of these nations.
It was usually kind of just a bonus, they got some troops from the other member states and occasionally demanded territories that they did not believe those states should control. Other than that there really was not much in that title, all of the regions in it were very autonomous.