They install into a subfolder of the Steam install folder. The mods folder in Documents is for mods you create or download manually. As long as they show up in game, it should be fine.

From my experience, steam will download all mods you subscribed. You can subscribe using your phone and the computer will download them. So i guess that also works in this case.
If you used local mods or UCP you will need to put them in manually again.

Thanks guys. I reinstalled the game again and resubscribed the mods again. All is working again. Thanks for you help.