My computer connected to Internet through a proxy. Is there any way to make updater to work? It just says - can't connect... Will the game work through proxy at all?!?

Unfortunately, no. Not yet at least.

Could you elaborate a little bit on your situation? What operating system, and what kind of proxy (SOCKS?)

There are probably some hacks you can do under Linux to make it work but I don't know if there's another way on any other platform. There may be easy ways for us under Windows to modify the client and updater to be aware of your proxy settings. On MacOS X, I have no idea.

The game itself uses a UDP connection, which will probably work through a SOCKSv5 compliant proxy but I've never tried it myself.

It should work through any proxy if you can get that UDP port opened either by settings or through your IT staff.

Under Linux, you can use "runsocks" and under Windows you can use a program called "SocksCap". They should enable any program to work through a proxy, but I have not tried them myself.