I want to play Dota 2 with my friend. However his Internet connection is so slow that it would take him about 2 days to download it and then it will be throttled. I am thinking of transferring the version I downloaded per USB-Drive to his Computer and account. Since this is a free game anyways. Would this work or will there be problems?
Yes, you can copy steam game files from one computer to anther to avoid having to download them again.
Simply copy the game files from the 'common' folder of the steam library where they are installed. By default this will be something similar to:
C:\Program Files (x86)\Steam\steamapps\common\GAME NAME
On the second machine copy these files to it's 'common' folder, then install the game as usual. When the steam installation gets to the bit where it would normally download the game files it will display 'Discovering existing files' instead.
Once it has finished discovering the existing files you'll be able to launch the game.
If you follow these steps and steam starts to download the game (instead of discovering the existing files) that means the files haven't been copied to the correct 'common' folder.
This will work between different computers and accounts as long as the game is free, or as long as the new account owns/buys the game.