There's basically no way to answer these questions using real physics I'm afraid.
It definitely can't close it's mouth faster than the speed of light. Yes it would have a strong gravitational pull, almost definitely so strong that it would just collapse into a black hole and not be able to exist. If it weren't that dense, then it would basically just be a big diffuse gas cloud that couldn't do much anyway, it would basically have to be a proto galaxy to not collapse into one.
If it's using magic to exist, then anything is on the table.
And it would close its mouth as fast as a snapping turtle can, but at scale. That sucker would cause a gravitic ripple as its mighty jaws go so much faster than the speed of light that it creates hyperphotons that ricochet back and forth off of the current boundaries of the universe.
It can do this because it is a non-physical entity from the higher dimensions where time is considered passe, and only suitable for microentities that have no conception of true sapience, yet claim to be the highest intelligence around.
Simply put, this is nothing more than a shadow of the shadow of a multi-dimensional being that dwarfs gods. It devouring the galaxy is akin to breathing for us.
Look, nobody said we had to limit ourselves to science fact.
The speed of sound is commonly used as "the speed of sound in the atmosphere", so something around 1200 Km/h.
Either way if a creature like that existed, you'd have to question what we are looking at, it's either NOT a galaxy in space, maybe a bunch of luminescent pollens or insects, or, if we are indeed in space, this would be definitive proof that there's some intelligent creator out there and the laws of the universe are clearly arbitrary.
It would still be unclear why such a specific silly event would take place, so I guess we'd have to give much more credit to the simulation theory.
So I guess it'd close its mouth faster than the speed of light.
I guess though, that if it was a single connected body, the black hole it collapsed into would be a much larger black hole due to the amount of matter and absolutely still a threat. Defeats the purpose of the image but still a hum dinger.
to reply to the question posed within the meme itself:
using the same physics which allow the creature to exist, i would create a device that consists of reflecting and refracting matter, like prisms, echo chambers, mirrors and lenses, with the overall capability to reflect and project a distorted 'circus mirror' holograph version of the creature at a subatomic level, which appears to be an order of magnitude larger, directly behind it.
then, when it turns around to face its new illusory foe, i would tie its shoelaces together.
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.
The speed of light is pretty slow at that scale, it would take thousands of years until the mouth is closed.
The speed of sound (in air?) is much much slower than the speed of light. See above.
How fast? I guess die, because that would cause a whole lot of other problems.
Yeah, the gravity would pull the galaxy to it, but it would also take quite some time (one normal galactic year is about 225 million earth years, we wouldn't instantly be pulled with an immense acceleration, we would probably just continue our orbit in a slightly different path or reverse, which would have it's effects but not immediate in our scale, right?)