If I install all the mods I have on one computer on another, would I be able to safely move my saves from one pc to another?

Without ever having actually done it myself, I'd say yes.

Yes, that's possible.
If you use mods, you have to copy them too or you will loose the things contained in the mods.

The saves tab in Wrye Bash is useful here to check you have the necessary mods installed for a particular save.