Makeup days from heavy September rains, that the school decided not to do at the end of the school year?
Apparently in the 60s and prior, some schools would have class on Christmas Eve - usually just a half day - with Christmas Day itself as the start of winter vacation.
How did Hinkle catch up to Frosty and the kids at the train station? For that matter, how'd he know they'd even be there?
He was bound to notice his hat was missing sooner or later (Hocus Pocus did replace it with a wreath after all). Plus, Frosty and the kids did make a pretty big scene when parading through town, and nobody wouldn't have noticed a talking snowman walking in the middle of the street. If anything, the townsfolk would have been talking about it nonstop, and an already incensed Hinkle would have learned about what happened to his prized hat. Since the weather was starting to get warmer, and since the kids were scene taking Frosty up towards the train station, he would have put two and two together and ran after them. It just took him longer to realize this seeing as he is a major idiot.
If Frosty doesn't know about things like traffic lights or how to count, how does he know that Karen sneezing means she's cold?
He is made of a cold substance. He's an idiot at this point, but he does know enough to at least get by alright.
Why did Mr. Twitchell decide to call his spray-the-snow-away product "Summer Wheeze"?
Maybe it made that sound at some point? Or he thought it would be a 'wheeze to use'? Or a play off of "Summer breeze"?
He probably thought it would sound more memorable. He was determined to be remembered for his product, and some other name might not have stuck out as much as he would have liked it to. Either that or some poor sap in marketing got worked to the hilt to come up with it.
How can Frosty now live without his top hat? In earlier specials, whenever his hat was removed he became as an ordinary inanimate snowman. Yet in this special, he can take his hat off and nothing happens to him.
A simple Continuity Snarl perhaps, but let's not forget that Crystal's kiss of love allowed Frosty to live without the hat. Then again, the next sequel after that did revert him right back to old habits, so perhaps it was temporary.