Many remember the scene where Professor Hinkle is told by Santa that he has to write an apology "a hundred zillion times" to receive any goods, but the dialogue originally began as a threat that if he touches the hat he'll never recieve another Christmas present again. Hinkle then names examples of things he would want. If there was continuity to the discussion, after he said that he had to make a living too, Santa would have said something like "Well you'd better not touch that hat, then".
The order to write the apology could have been a test to see how repentant Professor Hinkle was, or how seriously he took Santa's words. Even if he didn't write that many, if Santa looked in and saw him doing the job, it would mean Hinkle took what he said to heart.
Besides, real life conversations don't always have perfect continuity.
In Rudolph and Frosty's Christmas in July, Frosty and Crystal have children! Um...how do snow people procreate? Did they make them and bring them to life? Magic? Um...
That seems likely, as that's how Crystal and the snow parson who married them were created.
As this review says, "Thumpity thump thump, thumpity thump don't ask how Frosty mates."
Why does Frosty always say "Happy Birthday" when he gains consciousness? Whose birthday does he think it is? He can't be referring to himself, you don't say "happy birthday" when it's your own birthday!
When Karen and Frosty are locked in the greenhouse, Frosty melts and we cut away to have a look at Santa Claus during that time. But it takes a while to realize how truly traumatizing it would be for Karen to watch a freaking sentient snowman melt before her very eyes.
At the end when Santa dropped Karen off at her house, he left her on the snowy roof where she could easily slip and fall to her death.
It is possible that Karen had follow Santa's technique on how to get to her house carefully which is to walk carefully and go into the chimney.