Does anyone know where the drones originated and what galaxy they hailed from?
The berzerker drones were the product of an unknown and unnamed civilisation. There's no specific confirmation of who designed them, but given that they seem to lack intergalactic FTL, it's highly likely that their origin was within the present galaxy.
Out-of-universe, Joseph Malozzi (on his blog) mentioned that they'd mooted the idea of having the drone builders turn out to be the crew themselves, via their time-travel descendants on the Novus Colony.
Q: “I wanted to ask, was there any talk of who created the Berserker drones?”
JM: If you’re asking whether we considered the possibility that the crew’s descendants were responsible for creating the drones – yes, that was one possibility floated.
The creators of the drones were not specified or named; however, we do know that they were the enemy of the Ursini. It also becomes apparent that even after the war was won, the drones would keep fighting, attacking any foreign technology. It seems that they became a menace to all life in the galaxy because of their heuristic and adaptive programming which automatically targets anything "other" (presumably anything lacking a "friend" signature/signal of some kind). While they do seem to go into a hibernation-like state when/if all targets are neutralized and they are not summoned elsewhere, they are automatically reactivated by the presence or disturbance of any technology. This is presumably why they didn't wipe out the human "descendent" colonists—they used the Stargates to travel but had not been using spaceships (until the eminent disaster on their planet) to awaken the dormant drones. And if they had created them, that would have made humans the enemy of the Ursini in which case Col Telford would have undoubtedly be treated differently by the remaining Ursini, and they would not have destroyed the ghost town planet visited in Blockade, or the planet where they first discovered their descendants. And it's highly unlikely that they would have neglected to include programming to protect the Stargates from attack and damage from the drones since they relied on them so much.
There is no reason to believe that the drones originated from another galaxy, in fact the chances are extremely unlikely. The resources and technology required to travel between galaxies is another whole level beyond interstellar travel and technology. It's actually surprising and unplausible that the clear-blue aliens who captures Chloe would be capable and/or willing to chase the Destiny across a galactic divide.