Is there an easy way to integrate mods into games without replacing/modifying existing game files?
Ususally when a mod is installed it overwrites game files, which unless you back your game up can't be replaced.
Is there a way to easily add and remove mods?
Would it have to be a feature in like a game launcher?

it all depends on game and who makes mods. morrowind and oblivion mods are mostly hassle free.
HL2 mods as well.
both games are very mod friendly.
not so friendly games require more tweaking so you gotta replace some files.
and it also depends on mod creator. for example Fallout 2 mod which adds cut stuff comes with own installer. It replaces stuff by itself without user doing a thing.

Many modern games have mod managers where you can select the mods you want to enable. Sadly the only way you could make one for multiple games is to hardcodce everything, moving & swapping files around with a script

UT had a great mod control panel, you would select mods and it would switch for you. Source mods are treated like its own game, but these things have to be made by the devs themselves.