I would use boots regardless of showing or not, high jumps or not just for extra protection.

Use open front jump boots, as they allow the horse to feel when it knocks a jump.

It depends what your horse is like. I use boots all the time because my horse has very thoroughbred type legs. He managed to brush himself and have to have his leg hosed just from doing a 4 1/2 minute dressage test. (he had been wearing boots while warming up)

You can buy some quite cheap boots, there is lots on Ebay for about 5 pounds.

Yes please!!  No matter what height, accidents can happen.  I use open front tendon boots on mine to protect the tendons from being damaged either by poles or by the back feet getting too close to the front legs and damaging anything.  They're economical to buy with lots of choice so better safe than sorry.