I think I solved this problem and I believe it could help to thousands of other people desperately try getting Facetime to work on their machines (that is when it comes to Mac, still not sure what is causing iPhone/iPad Facetime problem some people are having).
Anyway... this is what I did and it solved it immediately. I am having this issue with Facetime on my iMac for the last two months and no solution provided on the internet in various forums helped a bit. Changing DNS servers, erasing com.apple.imagent file from the user Library/LaunchAgents folder, even adding loads of ports to port forwarding section of my router... nothing.
Simple solution worked for me! I'm thinking what's so different from my home iMac and my office one (BTW, exactly the same configuration, and almost the same applications installed).
Problem was in my PeerGuardian settings. I have it installed on both machines, but only one option was different. On my office machine, option was checked and Facetime worked without any problems. On my home machine this same option was unchecked and Facetime was never ever able to connect. After I checked this option (see link for screenshot), Little Snitch immediately popped up asking me to allow "applepushserviced" to connect. I allowed it and Facetime connected in a matter of seconds.
This is the setting in PeerGuardian you need to have checked so Facetime is able to connect.
Trust me, even when PeerGuardian is not running, this setting if unchecked will screw your Facetime login.
And this is what LittleSnitch popped up with after I checked this option and tried signing in with my Facetime again. I allowed it and Facetime connected. Problem solved, at least on my end.