You'll definitely have fun with egg and spoon! Me and my gaited horse win that one every time.
As for the other events, you can enter them with your gaited horse. I show in them all the time with my fox trotter (at a local, state, and regional level). If you are a good rider and you have a nice versatile gaited horse you should have no problem placing. But, I will say this (based on a lot of experience showing a gaited horse) you may not place as high with a quarter horse judge. If you enter a western pleasure class with AQHA rules and a judge from that background do not feel bad if you don't place very high. Some judges will judge a gaited horse the same as a non gaited one, but there are those judges who will not.
Also, if you want to do hunt-seat it is not normal to post on a gaited horse. Depending on the gait the horse does, you could mess up a gait posting or confuse the horse. So in hunt seat you will not probably do as well, if posting is required in the class. You may like Gaited and western dressage; they becoming more populare. Both of these events are very accommodating to gaited horses. Trail classes are also great for gaited horses.
Depending on the breed you get, you may want to check out their showbills for their regional and national shows. Breeds like the fox trotter offer several versatility classes including jumping, hunt seat, reining, ranch horse, western pleasure, and more.