Hi A.,
It sounds like your little girl is getting a lot of attention for not having the manners you would like to see. This in itself is very exciting to a three year old child, and it's apparent that she's learning how to get you engaged at a time when you would normally be taking care of yourself by enjoying your meal.
If it were me, (and it has been me in the past as a nanny), here's what I would do. First of all, decide what is most disruptive. In my opinion, it would be the bouncing around. So, when that happens, simply explain "You may sit at the table and eat with us, or you may play in your room." No time outs, which also keep you engaged, no threats of taking food away. Just give her two positive choices "You may do "x" or you may do "y", and when she bounces around again, just pick her up and take her to her room without a lot of conversation. "It's time to eat at the table. Come back when you are ready to eat."
At this point, leave it alone. Let her come back when she's ready, and let her eat. Don't discuss this with her, or you'll sabotage yourself. If her food is cold, oh well. The point is that you aren't engaging with her about this, there's no power struggle to punish her for not sitting still, and you go on about your business.
As for the chewing with her mouth open, I'd ignore it for a while, and try to correct it later. Some people just simply forget. It is, however, a great way to get your attention.
The point in all of this is to give her no attention for doing these things that are bothersome. Chances are, if you consistently redirect her to her room, after a while there won't be any fun in acting up.
Good luck!