Steam – If I download Steam on Ubuntu, does that mean I can play any game I get on Steam ON Ubuntu

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?

Yes and no. While Steam doesn't support this directly that I know of (though I rarely use Big Picture mode), you can use a symbolic link to make Windows "lie" to Steam. This will require a keyboard and/or mouse for the initial setup, but shouldn't require them after that.

Assuming your SSD's Steam install is at C:\Program Files (x86)\Steam, and you want your games to be stored at F:\SteamApps:

1. Make sure there's not already a folder at F:\SteamApps (or choose a different path).
2. Move the folder C:\Program Files (x86)\Steam\SteamApps to F:\SteamApps. At this point, no SteamApps folder should exist in your SSD's Steam folder.
3. Open a command prompt and run mklink /J "C:\Program Files (x86)\Steam\SteamApps" "F:\SteamApps"

Afterwards, if you look in C:\Program Files (x86)\Steam, you should see a shortcut leading to your HDD's SteamApps folder. When Steam queries for this folder, Windows will seamlessly send it through the symbolic link.

You can be more specific with your symlink, and only move certain games over, or everything under SteamApps\common, etc., but for the purposes of Big Picture mode, this should be your one-time-only setup step.