The soundbar is designed to be a replacement - rather than used at the same time - for your television speakers. The sound delay is due to the audio processing in the soundbar, so the delay would be considered "normal", since it should still be in sync with the picture.
The mute for the television controls only the television speakers. So the output jacks - RCA and optical - remain unaffected by the television being set to mute. So in essence, you would have to use the speaker bar remote to mute the speaker bar.
It may not be the answer you prefer, however that was the design and this is considered normal functionality. There's not a way I know of that allows you to control the soundbar mute with the television volume controls.
I've noticed that when I have both TV and sound bar speakers on, there's a < 1 second delay (sound bar via optical output). Is this normal / fixable? Second, if I set my tv to external speakers, I'd still expect that mute would work as expected. But I have to use my speaker bar remote to mute those speakers. Is this normal as well?