Can you remove a weapon mod from one item, and place it on another item?

Only if you replace them with another mod.

I save all my standard mods to use as swap-outs to get higher valued ones from gear that drops for me.

Yes and no.

Mods from one class of weapon are not swappable to different class.
E.g. Large magazine from 10mm pistol can go on any other 10mm pistol you find. Or say you found crappy 10mm but it has a better automatic fire receiver - this can be taken out and put onto any other 10mm pistol

But any 10mm mod wont swap to pipe pistol.

To swap mods, you also generally have to replace it before taking it out (some slots like muzzle are free to detach). Just save at least one low level mod for each part slot for each weapom class, and you can then swap better parts out and replace with the saved crappy one.

yes but you have to swap them out w/ another mod 1st if its a good mod like a large fast eject mag it's cheaper to spend the adhesive on a stock mag & bank the modded mag