Hi, a little technical question!

Is it possible to somehow play half-life 2 through it with friends? After all, all the tools in it already exist?

Yes there is a mod for garrysmod with this mod you can load the half life 2 maps and then play it with your friend or just play synergy its free on steam.

Yeah! Check that.

I believe there's got to be mods that allow you to do that yes. And who knows, maybe Valve will release an official co-op campaign like they did with Portal 2

Search game in steam with name "Synergy", its co-op Half-Life 2, and "Sven" its Half-Life 1 co-op