What if extraterrestrials wanted to take the moon? (hypothetical)?

"there culture" - the word there means in a specific location. Are you suggesting their culture is location-dependent?

Take how?

As long as they don't take it out of its orbit they can take their vacations on the moon as much as they like.

If they would remove it it would remove the periods of all our fertile females and the monthly rhythms of many other earth-creatures, as well as mess up our tides

Seems to me we shouldn't respond at all.  The only people really inconvenienced would be the ones who get their electricity from tidal power, and they aren't very numerous.  (I also think of the pollinators of night-blooming cereus, but I'm not sure if they actually depend on moonlight.)

If the aliens are capable of taking the moon, it seems unlikely we would have the power to stop them, so best not be provocative.  However it might not be unreasonable to ask them for something in trade -- for example a replacement for tidal power.

Let's entertain the unlikely hypothesis that ET actually exists:

Why do you think ET would want to "take the moon". Just because ET is up in space, the moon is up in space, therefore ET would be mucking about with the moon? That is strange reasoning.

We know the moon is a desolate ball of rock. Nothing there to attract anyone, not even us anymore. So ET has travelled a minimum of hundreds of trillions of miles, and what do they do? Hang around a ball of rock. What would YOU do if you had the magical mythical ability to travel anywhere in the universe?

If they were able to 'collect' the moon, I'm sure we'd say, "Use it in good health!" as they flew away...

Do you have any idea how much force it would take to move the moon from orbit? I don't either but it is a LOT. 

Some of us would shake our fists and fire our guns in the air to discourage them.   Some of us would roll over and say that we will miss the moonlit nights.  Some of us would expect Space Force to take action.  Some of us would just be thankful that there are ETs advanced enough to collect moons for a hobby.

Some of the answers are here in the link:

https://insh.world/science/what-if-we-suddenly-los...

If they wanted to take the Moon away they'd be so much more advanced than we are technologically that it would not be possible to stop them.

Why would they need it really and what could we do about it?

 It lacks Water and Minerals

The only possible use it could have is Ballistic