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?
Run as Administrator
First, completely exit Steam, and right-click on the shortcut which you use to launch Steam and select "Run as Administrator." This step may be enough to fix the problem. If not, continue below:
Additional Installers
You may need to manually run secondary installers required by this game. Exit Steam entirely. Browse to your Grand Theft Auto 4 installation folder (Usually C:\Program Files\Steam\steamapps\common\grand theft auto iv\)
You will need to run each of the following installers:
...\steamapps\common\grand theft auto iv\Installers\DirectX_jun2008\DXSETUP
...\steamapps\common\grand theft auto iv\Installers\DirectX_Mar2009\DXSETUP
...\steamapps\common\grand theft auto iv\Installers\vcredist_x86
Games for Windows Live
Also, install the latest version of the Games For Windows Live software, which can be found here: http://www.microsoft.com/games/en-US/Live/Pages/livemarketplace.aspx
Restart your computer and test the issue again. If you continue to have difficulty, run this installer:
...\steamapps\common\grand theft auto iv\GTAIV\Activation\SteamActivation
References
Taken from this steam forum post. In addition, there is another thread of a huge list of GTA IV steam issues which you can find here.
Best Answer
Once you enter the key the game should be added to your Steam library immediately. The dialogue you entered the key should have given you the option to install the game. If you missed that or chose not to install the game then you can install it by selecting the game in your library and clicking the blue install button.