There's not *just* cross country competitions unfortunately. Sometimes you can find derby type shows with stadium and cross country combined into one course, I've done a few of those and they are a lot of fun. Two-phase or combined training shows are fun too, you do dressage and stadium and they combine the scores to find the winner, so it's more of an eventing format.
Shoes will help if his feet are a little tender barefoot, but keep in mind when you put shoes on you will be taking away the natural grip their hooves provide. If you are just starting out and doing small jumps it probably won't be an issue, but just something to be aware of. If you're riding on hard ground or in wet grass he may find it a bit more slippery with shoes on than if he was barefoot.
Some hesitation is natural when a horse is jumping a new jump or trying new things. The best thing you can do for him is be really quiet and supportive with your riding and body position. Eyes up, shoulders back, leg on, just really positive riding, your trainer will help you with all of that. If you're a confident rider, your horse will become more confident. And it's very common for horses to look or refuse at new or different jumps or courses. That's why the riding positive is so important, so the horse learns that it might be new or different, but they jump anyways.
It's really important to just take your time introducing all the cross country jumps and questions, always going with your trainer and having a schoolmaster to help lead your horse over new things helps a ton. It will just take time, but it's a ton of fun regardless of what size jumps you are jumping!