No, the use of custom roms (CM11) is not recommended by the authors, and they do not support their app running on custom roms.
The only solution to the issue would be to use a stock rom, I remember CM10 being mentioned in this thread a while ago, that rom should work OK.
Actually, Facebook does not play videos without rebooting at some point.
They start out stuttering and at the end the pone reboots.
That's indeed an old and well known issue that still hasn't been fixed.
I'm having the issue for months with CM11 but hope for a solution at some point in time.
Dlna works, tested on LG smart tv. I used some app from the store.
Voice recognition works ok, but I did not try to download packages for offline speech recognition.
For those on the custom kernel (Chill): Unstable operation may be a result from the standard overclock when using a custom kernel.
Not all chips are binned equally, some can get unstable when overclocked.