When it comes to the mouth being exposed. I would personally say, it would no doubt be the same reason many army/swat helmets don't cover the mouths. When it comes down to it, the axial skeleton is really what needs to be protected. Same deal with the Captain America costume. This also allows for various rebreathers to be applied with little fuss unless of course the suit already has it's own air supply. It's a different story for robin and the rest. Secondly as I said before, covering batman's mouth really wouldn't be met well by people/fans/trades/faraci..etc. Same as having a batman without a cape. Not without in context explanation. You have both functional and aesthetic rhetoric there.
The belt being yellow/gold...I've yet to hear any such thing.
What's more I've seen plenty of black or non yellow belts in continuity before so you can do this thing and leave none the wiser, as opposed to a cape free batsuit.