Here are some comments on all of them. All 'off the top of my head' without checking anything:
baking contest
Yes, why not. I guess you could call 'The Great British Bake Off' a baking contest.
middle school
Depends on the local education system. There are some 'middle schools' (ages approx 9/10 to mid teens) in the UK, but not many.
exercise regimen
Not sure I'd say 'regimen' . Perhaps 'exercise regime'? or 'exercise routine'?
T-shirt cannon
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.
graduate school
No, this is part of the American system.
sweatshirt
Yes, this is the standard term for a long sleeved cotton top with a slightly fleecy lining.
laundry cart
No, I don't think so. This use of 'cart' sounds American. Not sure exactly what it is.
bake sale
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.
semester
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.
speedometer
Yes.
baking pan
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.
term
See above