I found words such as "колядка" and "щедривочка" -- are they correct?
"щедривочка" is ukranian term. I do not know many about it.
As for "колядка" - it is common slavic term (unkranian too).
First of all "коляда" is slavic word for celebrations of pre-Christmas, Christmas and post-Christmas periods.
On of the most famous rituals of these periods is "колядование". During "колядование" people (usually chindren) visit random houses and sing "blessing" songs in exchange for gifts (usuallt sweets and candies). These songs are called "колядки" or "колядные песни".
I cannot understand from wiki is "carol" for "колядки" or more common "рождественские/новогодние песни" (which may be something not binded to ritual of "колядование", for example famous "в лесу родилась ёлочка...").
But your song sounds exactly as "колядка" ("...По всем домам мы обходим...").

"Выглядя так приятно!" sounds unnatural in this context.
I think "Такие нарядные!" will be much better (something like "so good-looking").

Rest of the song sounds good. However "Под листьями, такими зелёными!" sounds STRANGE in russian winter.
P.S.
Maybe "Among spurces so green" will be better. Russians never call needles of ever-green spruce as "leaves/листья" ("needles/иголки" is used instead).