Only if the moon is up.

Imagine seeing the moon just switch off

Yup

If you can see the moon (if it is "up" at night).

Kind of. The concept of simultaneity breaks down at distances where the speed of light matters. If we base it on what we currently observe and call "now" on the Sun the eight minute old state we currently observe then what does "now" on earth look like from the point of view of the Sun at that same moment? You can't reconcile a single "now" for observers in both locations.

An alternative take which is also consistent with observable physics is that the speed of light is infinite but it's causality itself that propagates at c.

Thinking in those terms also makes a number of relativistic effects more intuitive. You need infinite energy to reach the speed of light simply because it's infinitely fast. Time dilates when moving because you're encountering approaching causality earlier than you otherwise would have. Time "stops" for anything traveling at the speed of light because at infinite speed it just experiences literally everything in its line of travel at once and the concept of "after" becomes meaningless, encountering all future oncoming causality in a single instant.

This was a bit of a tangent but it's something that has fascinated me for a long time.

It's sort of how if you hold a slinky on one end hanging down, then drop the slinky, bottom will not start falling until the top reaches it. In a sense, bottom will be hanging onto nothing. But of course that nothing is tension from the top of the slinky.

That is correct as weird as it sounds

The sun could be gone but its influence would remain. Kinda like getting out of a pool and looking back to see the waves on the surface that you caused.

Depending on the lunar cycle, the night time side would notice the moon become dim.

Moon would "disappear" when it no longer reflected Sun's light.

It would also start getting very cold fast

It takes 8 minutes for the light to travel from the sun to Earth. Because light in a vacuum travels faster than anything, including information, we would not and could not know it had disappeared for 8 minutes. This means Earth would continue to follow its orbit around a non-existent sun for 8 minutes because the Sun’s gravity would still be acting on the Earth.

If it was nighttime, you wouldn’t notice the sudden lack of sunlight (other than if it was a full moon) but you’d almost certainly notice the change in gravity.

Edit: actually, you wouldn’t feel any difference in gravity or experience any change of acceleration. What you would experience is a very tiny vibration, of 1 million push notifications being sent to your phone from the other side of the planet.

Yes, because of the medium of communication you are using right now.

The classic sci-fi short story A Pail of Air touches on this.

https://www.gutenberg.org/ebooks/51461

Not anymore…

Which two event are you talking about being simultaneous? The Sun going out and Earthers observing it? Those things will not be simultaneous in any reference frame, because they are "light-like" separated. (ie they lie on a 45 degree line in a Minkowski plot.)