...stationary orbit around the earth? Obviously, if possible, a lunar stationary satelite over the "dark" side of the moon would be undetectable from earth. This could be useful for certain purposes.....

It's possible. To have a lunar geostationary satellite on the dark side you'd need another satellite that could "see" Earth and the satellite on the dark side otherwise the dark side lunar base would never be able to receive Champions League football.

No reason why not - The Moon has gravity - albeit less than the earth - and that is what is needed for stationary orbit.

I can't think of a purpose for this though... and how would you get the signals back to earth?.... the moon would be in the way.

Yes. It would be a very good place to stick a satellite telescope. no distortions etc.

I think the Justice League already has one up there.