Steam – ny way to run an older version of the Steam client

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?

Age of Empires Online is a free-to-play game with optional pay-to-play DLC. The All in Wonder bundle is a bundle of this DLC for the base game. This might seem like a minor distinction, but it's an important to understand what's happening here.

As with other free-to-play games in Steam, the base Age of Empires Online game will not appear in the "All games" list like a normal purchased game would, even if have purchased DLC for it. However, if you go to the steam store and search for the game there, you will be able to install the game client through Steam as normal.

As your paid-for DLC is associated with your Steam account it should still be available when you log into AoE Online through Steam.