The article I linked to you to is a problem that is hard to fix by updating the flirc.
Basically, MCE/XBOX/Microsoft remotes alternate between 2 different "signals" when you press the same button.
So the solution in the support article aims to capture both signals. You may have to do this a few times for it to work.
If "redo" window comes up then flirc has already recorded that signal. Keep trying to record then it should pick up the other signal, it will say "recorded sucessfully".
This needs to be done for all keys.
The "every second key press doesn't work" issue doesn't usually indicate that Flirc is broken - it's a quirky feature of some older designs of remote.
The issues with the IR receiver on the TV are strange and are outside my knowledge (I just try and troubleshoot common issues), Contact Jason on this link and see if he has an ideas as he is more knowledgeable than I:
http://flirc.tv/contact