"One cultural translation mistake: "It's come to us for Christmas". The correct translation is "It has come to us for the celebration". In Russia the tree and presents are for New Year's Eve."
Since you are being a stickler for correctness, I must point out a literal translation would be 'holiday' without specifics.

A (little) fir tree in this case would be acceptable since the tree was born (little) and grew up in the forest. None are born at their full growth.
That adjective is then understood and commonly used in tales for children.

"Under the runner's tread" might better read "Beneath the runner's track(s)".

(As to your follow up below) ...
Трусишка зайка серенький - perhaps, "The timid/cowardly/fearful/scared grey bunny"

"Finally, I would use "sleigh" instead of "sledge". In America, the latter word is usually associated with a big hammer aka sledge (hammer)."
That statement is false. Throughout the world when horsepower was the dominating power driver a sledge was a heavy-duty transport vehicle.
As you may note in the lyrics firewood is being transported. Sleigh generally referred to light duty vehicle that usually transported people
and/or lighter loads. A sledgehammer is a 'heavy' hammer as opposed to a regular duty or light duty hammer.

As to this:
https://lyricstranslate.com/en/v-lesu-rodilas-yolochka-forest-fir-was-bo...
Look to your laurels... there is opportunity for improvement.

Do not address others in such a manner (especially during the holidays). It is very bad manners on a site meant for worldly understanding.
Should you feel the need to reconstruct your 'suggestions' in a more amical manner let me know and I will consider my need to address this issue as the past and it will fade in time.

Huh.... Maybe he really did leave... But I'm sure he will be checking in every now and again to find out if his 'requests' had been 'answered' 😮‍💨