Yes, especially if you exceed the allowed bandwidth. The cost is usually quite high.
You wouldn't be running up their "Wi-Fi bill", but you would be running up their Internet bill if you consume content such as videos, music, and large downloads while connected. Especially if the Internet connection is metered or capped, like keerok said.
Commonsense. IF the wifi account is metered, and you overshot the allowable bandwidth for that billing period then yes, the bill will go up.
No. Wifi is run from the router. You might mean megabit usage. in that case yes it is possible if they have limited bandwidth usage. After they reach their limit internet companies charge per meg.
Yes if you are watching any sort of movie, this consumes bandwidth/data, if the wifi is on a datacap contract a certain amount of data usage is included, if the data limits are exceeded then the user is billed extra
who knows but that had better be your wifi or you are stealing