Full pelt gallop: Two reasons: one, was to give the kid a real show to entertain him. Like soldiers in parades saluting kids, or truck drivers honking when kids make the "blow your horn" gesture, it's a classic example of heroes showing off for kids who admire them. Second reason, the postman rode past the house and missed the mail. As a way of showing how devoted the postal service was to its mission, the postman doubled back as fast as possible to pick up the letter. If he had walked back, it would show he cared less.
Think about it. You wander off after borrowing a pen from someone. If it's no biggie to you or they aren't someone you care about impressing, you toss it back to them. If its someone who's opinion you really value, you run back and hand it to them personally.
Candlelight: You need gas to run generators to get power (unless you live on a hydroelectric dam like tom petty.) There aren't any more oil refineries, so gas is precious. That's why everyone rides horses. But on very special occasions (like hearing that there's a Restored USA, a president, and a mail service) you bust out your supply and start up the generator for a good old fashioned town dance. Think of it like opening a bottle of good champagne on a special occasion.
Not to say there aren't issues with this movie (zippos don't hold their fuel for more than a few weeks, jackets on decaying bodies soak up all of the putrefaction, and many more), but this is NOT as bad or illogical a movie as furious hollywood execs and faddish critics would have had you believe in the late 1990s.
The Postman put his life in danger and he knew he was adored by the people who had heard of him. So why shouldn't he be allowed some theatrics? If he had just had his horse trotting back slowly to get the letter, very likely the Letter Boy (in reality played by Costner's son) wouldn't remember him so well and certainly no statue would be made of him and we see the grown up Letter Boy exclaiming "that was me"!
Because he doesn't want to let the kid down and it means a lot to the kid. It's not just a postal service to the kid or to them, it's trying to bring back civilization. He's doing what he needs to do and possibly ignores the kid as he has much more other things on his mind or might be at the verge of giving up, the kid might have even given him determination to continue, so he gallops to retrieve that letter.
As for the party... It's a party?
If you weren't too busy laughing, you might have not missed these OBVIOUS things.
Watch the scene again. It shows The Postman with a pissed-off look in his face due to the insolence of this kid who was tardy in having his letter ready. So he turned back around and tried to trample him at full gallop but then had a pang of conscience at the end and grabbed the letter instead.