Hi Robin - Bristish English answer here for you.
In the first question, bill is the right word (not order) but you would phrase it differently. You could say: "I'm going to go and pay the bill", or "I am going to settle up" or "I am going to go and pay".
With the second situation, although I have heard people say "foot the bill", it more commonly is said about someone else, than a self-declaration. For example, "At our wedding, my Dad is going to foot the bill for the all of the wine".
If you wanted to express that you were going to pay food for a group of people, you would colloquially say, either: "I've got this", or "It's on me", or "It's my treat".
I hope that this helps.
Tom