You can barrel race any breed of horse who is willing to run around three barrels😊 Turning may be harder but you'll have longer strides to compensate!
Yes u can barrel race thoroughbred my friend has a Rv and she is pretty successful ☺️
Not well or successfully. TBs are bred for and run better in longer races, whereas a breed like a Quarter Horse is meant for shorter races. (That's actually how QHs got their name--running quarter mile races.) TBs are very athletic and willing, though, so I would suggest something your horse will excel in, like jumping, hunters, eventing, etc.
I wouldn't. They just don't have the conformation for it. Do something more fun like eventing