That's a very ambitious training program, when is the show?
The best way to use the side reins is on a lunge line and with supervision. You need someone with you to show you the moments when he is working correctly and how to encourage them. I would NOT recommend using them while riding.
The only practical reason to use a standing martingale is if the horse slings his head or raises their head above the effective range of the bit as an evasion. The irony there is that a horse that does either of those things won't pin in the hunter ring.
They are popular mostly as a fashion statement in the hunter ring. If you want to put it on because it looks "huntery", go ahead, just make sure it's adjusted correctly. Use the D-ring instead of the loose ring for the same reason, the loose ring looks more dressagey.
Your plan for gymnastics is fine.
Your plan for teaching flying changees is worrisome. That's a tall order, especially if you only have a short period of time. Teaching flying changes over a pole can be dicey as well. My best advice is to really work on improving the quality of his canter, making it balanced, round, rhythmic and three beat. Don't even think about attempting changes until the pony holds a good balanced canter with minimal support from you. The best short term solution is to teach him to land from a fence on the lead you want.
And if you're doing a handy course, practice trot fences and moving up and coming back from a hand gallop, handy courses often include them.
The very best thing to work is a steady, consistent pace with consistent distances. That's primarily what the judges are looking for (after manners and suitablity.) Figure out the pace that gets your pony to his fences most of the time at the correct balance and distance, then learn to recognize it and create it on demand.