To answer the OPs question - It really depends on the horse. You have to look at the horse as an individual & see how his/her mentality fits into the mold of the ideal horse you are looking for. Some show horses will have a hard time out in the woods. Some trail horses will have a hard time with the demands of the show ring. Many other horses are quite adaptable regardless of what their current job is.
I'm another person that takes my show horses out on the trails, and I've also turned around trail horses to be show horses. A good mind is the key to versatility!
PS...are you looking to buy a horse for your daughter? I know a great gelding that would fit the bill.