No. This would move parts of the military that already do space stuff (primarily Air Force) into their own branch. The OST doesn't ban all military things in space (GPS constellation for example, is totally legal). The OST does ban nuclear weapons and 'boots on the ground' when it comes to certain extraterrestrial environs. But again, despite the name this isn't a Space Marine type organization that is being proposed. Just a reorg of existing capabilities.
Given its size and operating specs I'd say it's a space drone designed for superiority in the orbital range. It enables the Air Force to attack and defend at very high altitudes against otherwise unreachable targets and threats. Or, at least now, since it doesn't appear to carry or need any weapons systems, it can tow satellites out of orbit (for security reasons, American or foreign) or run very specific rescue missions.
No. The Outer Space Treaty forbids WMDs in orbit, but conventional weapons are fine. If the US decided to annex the Moon or put nukes into orbit, that'd be a violation.
(Incidentally, Soyuz capsules already carry firearms into space, in case of bears/wolves/etc. at the landing site.)
Not to mention the USSR put a 23mm autocannon into space onboard a space station.