Steam – How to send friends their parts of the Magicka Four Pack

Yes, you can do this. After playing the Steam-installed version of Frozen Synapse (so that you have a registered account and email address), you can go to Mode7's website and use the form there to request a link to download and install a copy of Frozen Synapse independently of Steam.

Note also that the Steam-installed copy can be run without going through Steam. Just browse into the install directory and run the exe directly. (Don't use the desktop icon created by Steam, as that runs it via a steam:// URL instead of starting the exe directly.)

I have personally used both of these methods, having initially installed FS on my dual-boot laptop via Steam, set up an icon to run it in Windows 7 without starting Steam, and also run it under Ubuntu without Steam even being installed. I then requested a download link from Mode7, used it to download the OS X version, and installed that on my MacBook Pro to run without Steam.

You do not need to re-purchase any game you've bought previously, as long as you log in using the same Steam account you previously bought the game with; however, you will need to download those games on the new computer.
It is possible that Steam will ask you to enter a code, sent to your e-mail address, when you attempt to log in using a new PC for the first time; this is a measure meant to protect accounts from being hacked and may be deactivated (not recommended).

My library is empty:

According to Steam Support, you are either logged into the wrong account, or your filter is not set to "All Games".

Please ensure that all games are visible by clicking View > All Games on the upper-left corner of the Library > Games list.

According to Steam Forum, it often happens that people accidentally create new accounts when trying to log into their old ones.

It appears that in some rare cases, programs may interfere with Steam.

How can I tell if the account I'm logged in with is my own?

• Click on [name]'s Account > Account Details located on the top right corner of Steam and check your Licenses + Subscriptions or your Store Transactions.
  You should see a list of all games you have purchased.

Should your library be empty, despite being logged in on the correct account, you may want to contact Steam Support directly.