Close its mouth at the speed of light?
Close its mouth at the speed of sound?
Close its mouth faster than the speed of light?
Also… wouldn’t it be heavier than the galaxy itself? So it would create its own gravitational pull that’s like extremely strong?
The speed of sound is the velocity a force can propagate in a medium, so yes even assuming such a mega structure is theoretically possible (it's not) it would close its mouth at the speed of sound.
If you want a galaxy-doom monster, either pick a super massive black hole, a super massime white hole, a super massive black hole coming out of a bigger white hole, or a super massive quasar pulsating high energy X-ray bursts with enough power to kill all life of an entire galaxy
That's not entirely accurate, the force it applies to close it's mouth would probably travel at or close to the speed of sound along its jaw, but it could reach the speed of light by applying that force if you ignore a number of problems. One of which is that it will turn into a black hole at that scale, it's much too dense.
On your second point, it's hard to make any of those into galaxy killers. Supermassive black holes exist at the centre of virtually every galaxy and don't do a ton, and even quasars only have limited killing range as there are limits to how collimated a beam of radiation can be. White holes are more complicated and I don't have enough space in a single comment to go into the nuances, but they're about as harmless as black holes really.
The only thing I can think of that would destroy all life in a galaxy would also destroy the universe, and that's to trigger a false vacuum decay, but that might not be possible anyway.