What extra steps do I need to take to play and patch Crysis 2 on the PC

Crysis Warhead isn't a multiplayer component. It's a fully single-player game in fact, telling the story from the viewpoint of Psycho, with in my humble opinion better gameplay than the original.

You're talking about Crysis Wars, which is a separate game bundled with Warhead, as its multiplayer component. This is in contrast with Crysis where the multiplayer was integrated.

I would say getting Warhead is definitely worth it to add to the story and have a multiplayer game.

Yes, you can trust it. It is non-intrusive and all it does (at minimum) is manage your games and patches for you. Anything else is optional.

Steam has an offline mode, where you can play your games without a connection to Steam, granting you total freedom from Valve's servers.

It doesn't install anything but the software you need (like DirectX and drivers for example) and the games and patches. Nothing you don't want, unless you don't want the game you're installing. The only downside is that these patches are kind of forced on you, even for single-player games. If the developer decides to 'downgrade' the game or make a change you don't like, there's not much you can do about it.

I'm not aware of any 'independent verification', but the DRM Steam is itself is very harmless, you can choose to run Steam in offline mode and there would be absolutely no way for them to pull access to your game, except for the fact you wouldn't get patches. Any other DRM can be chosen to be used by the game itself of course, but that has nothing to do with Steam.

Anyway, the choice to use Steam has already been made by Square Enix. If you don't trust them, why are you installing the game itself?