I'm not not sure what a "Jewish law perspective" means in this context, but I don't think it's reasonable to describe the song as "all Old Testament". Sure you can point to Isaiah, but you can also point to Matthew, and anybody singing it today is certainly identifying "Immanuel" with Jesus.
I agree with this. I'm Christian and happen to love this song, but I'm not so sure how to answer your question. It's an Advent carol, not a Christmas one, but it's Christian either way. Yes, it originates with the verse from Isaiah so you could try to characterize it as Old Testament and therefore not at odds with Judaism, but it is certainly also consonant with what's written in Matthew ("and they shall call him Emmanuel, which means God with us" in reference to Jesus) and it's definitely sung in that context in Christian churches.
It's a church song - so do church songs belong in a public school? It depends on what the purpose is - my kids have come home from school singing the dreidel song or with a picture they colored about Kwanzaa, I have zero issues with that if it's done in the context of exposure to/learning about other religions or cultures.
OP, I don't see the issue with Jingle Bells. That, to me, is like Frosty the Snowman or Winter Wonderland. Frankly, Rudolph the Red-Nosed Reindeer is more "Christmas-y" than Jingle Bells. Little Drummer Boy, however, is also a religious song.