Also, are there other military inclined monasteries in the Buddhist world that follow teachings similar to Shaolin? Or are they an exception like Sparta was in ancient Greece?
I can only talk briefly but...
The style of Buddhism practiced at the Shaolin temple is a type of Chan Buddhism. Chan Buddhism evolved from Mahayana Buddhism that traveled along the silk road before reaching China.
A core tenet of Chan Buddhism is hard physical labor to achieve enlightenment. By working your body to exhaustion, you can free your mind type of thinking.
When you read Hui Neng's Platform Sutra, he talks about how he pounded wheat into grain (or something similar...it's been a while) over the course of weeks, allowing him to reach a level of "knowingness."
It's not a far jump to make from hard physical labor -> martial arts, especially as the monks living in the early days of the Shaolin temple had only themselves to rely on to fight of bandits/raiders.
I can't link it now, but if you want me to I can give you a bunch of sources about the history and spread of Chan Buddhism.
This doesn't directly answer your question as relating to Shaolin monks, but gives some insight into Buddhism and violence https://www.reddit.com/r/AskHistorians/comments/3eg3w5/how_did_buddhistmajority_nations_reconcile_state/
Shaolin monks initially practiced martial arts as a way to defend themselves from intruders/bandits. They also use this as a part of their religious life in terms of gaining spiritual wisdom through physical training. Their training in the martial arts was not used as an aggressive stance, but rather as a defensive stance.
They still focus on the tenet that one should always attempt to resolve a conflict through peaceful means and should only resort to physical means as a last measure.