As for your first question, it depends on the era you're talking about. Prior to the Thirty Years War, most states did not have a standing army, but rather formed/hired an army as needed. However, this was not particularly unique within the HRE, given that standing armies were very expensive, and most states did not have a standing army. As standing professional armies became more common in the post-Westphalian period, many of the states did maintain standing forces.

In terms of more general autonomy, the various states of the HRE did have a degree of say in how/where troops were deployed. The Reichstag--the Imperial Diet--which consisted of the electors, princes, and free cities, could vote to raise troops or money for the defence of the Empire in general, and the Emperor could request such a vote for defence against a particular threat, such as the Ottomans advancing against the Empire's southern frontier, or the French in the west. Of course, there was a difference between the Reichstag voting for money/troops to be sent to the Emperor, and that money and troops actually materialising.

The Empire was subdivided into several 'Circles', which were regional gatherings of princes in certain regions of the Empire. The Circle was essentially an attempt to localise defence amongst the princes. Each Circle had its own Diet for deliberation amongst the princes that were apart of the Circle, and were meant to allow them to co-ordinate their responses to threats and collection of taxes voted for by the Reichstag. In times of Reichskrieg, the Circles were responsible for raising and supporting a set number of forces in order to form a combined Army of the Empire, independent and seperate from the army of the Emperor.

As for your second question, it's important to note that from ~1500 on, the Empire was under the rule of the Perpetual Public Peace, which obligated the princes of the Empire to settle their disputes through the Imperial legal system, rather than through force of arms. While the Thirty Years War serves as an obvious counter-example, the Public Peace was generally respected, and escalation to force of arms within the Empire relatively rare until the complete breakdown of the Thirty Years War. It does depend on how you define 'regularly', of course, as the post-Thirty Years War period through the whole of the 18th century had many wars, but whether or not this counts as 'regularly' fighting battles is up for debate. It is worth noting that in many of the major wars throughout the period, there was fighting in and amongst the princes of the Empire (See War of the Spanish Succession, War of the Austrian Succession), but in the Nine Years War and Great Turkish War, the Empire presented a relatively united front. The princes of the Empire did indeed fight against each other and foreign powers, but whether you can characterise that as 'regular' infighting or fighting against external powers is going to depend on what you want to focus on.

Hopefully this answers your question, and feel free to ask any followups!