Here are some comments on all of them. All 'off the top of my head' without checking anything:
Yes, why not. I guess you could call 'The Great British Bake Off' a baking contest.
Depends on the local education system. There are some 'middle schools' (ages approx 9/10 to mid teens) in the UK, but not many.
Not sure I'd say 'regimen' . Perhaps 'exercise regime'? or 'exercise routine'?
Sorry. No idea what that is.
prepaid parcel bag
Hmm...no. A prepaid bag you put a parcel in? I get the concept but don't think I've heard this term
rental car company
Do you mean 'car rental company'? If so, yes. We do say that in BrE.
No, this is part of the American system.
Yes, this is the standard term for a long sleeved cotton top with a slightly fleecy lining.
No, I don't think so. This use of 'cart' sounds American. Not sure exactly what it is.
This sounds very American. But we've all seen them on TV and in the movies, and it wouldn't surprise me if the term is starting to be used in the UK as well. We'd usually say 'cake sale', I think.
As with 'middle school', this depends on the education system. In general, the UK academic year is divided into three terms. However, if a particular academic institution happens to divide their year into two, we would call it a semester. This is about culture rather than language.
No. This is AmE. The equivalent in BrE is 'baking tin' or 'baking sheet' (if it's flat). In BrE, we only use 'pan' for things that you put on top of the stove.