Frosty's scarf was a gift from Santa Claus.
I think I know why Frosty wears a scarf in this movie (and in Rudolph and Frosty's Christmas in July) when he didn't have one in the original. Santa Claus probably gave it to him as a token of their friendship. Elsie, the redheaded girl, was supposed to have been Karen.
Since this special is clearly meant as a direct sequel to the original Frosty the Snowman, and even had the same writer, Romeo Muller, it seems strange that Frosty's best friend Karen should be missing. But she does have an Expy: Elsie, the girl who converses the most with Frosty in this special and who suggests the idea of building him a wife. Maybe in the original draft of the script for this special, Elsie was Karen. But it turned out that for some legal reason, Karen couldn't be used, so she was given a name-change and a redesign.