This is from my view point, definitely not an authority on the subject...It's a really difficult subject to answer and it's bound to cause divisions, but here is my attempt. I'm open to disagreement and correction, since this is a pretty deep discussion.
The Talmud is collection of writings by many Rabbi's from the past. Just like most Jews don't see the New Testament as scripture, The Talmud isn't viewed as scripture by most Christians. I do think some Jews are a little curious about the New Testament, just as Christians are a little curious with The Talmud. Obviously, they both have reservations on what they say.
Yes, I do see the Jews as God's chosen people. They were trusted in keeping the law and showing who the true God is. At the same time, Jew or gentile can be a Jew by trusting Jesus as Lord.
Galatians 3:29
And if you are Christ's, then you are Abraham's offspring, heirs according to promise.
Galatians 3:28
There is neither Jew nor Greek, there is neither slave nor free, there is no male and female, for you are all one in Christ Jesus.
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Now most Jews today see that there are two kinds of Jews, Jewish by blood, and Jewish by faith. Not gonna argue this, what's most important is your decision on Jesus.
John 14:6
Jesus answered, “I am the way and the truth and the life. No one comes to the Father except through me.
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This is where the great divide is in Christianity and the Jewish Faith. I see Christianity as a Jewish Faith, since Jesus was/is a Jew and was the greatest Rabbi ever. Where most Jewish Temple's reject Jesus as the Messiah due to difference of interruption of scripture, but most Jewish Temple's acknowledge The Talmud and Rabbinical Traditions. It doesn't make sense maybe, but it's a really complicated subject.
I have been to a few Jewish Temples (Orthodox, Conservative, and maybe Reform) and my experience is, they don't want to talk about Jesus, and they think Christians are Goyim. Karaite Judaism sort of rejects Rabbinical Judaism and is really only concerned with what the Bible (Torah) says. I'm pretty sure they reject The Talmud. I would love to attend a Karaite Jewish Temple, really think it would be a place of learning. I watched a few YouTube channels and their view points are refreshing, but be warned they don't typically believe in Jesus as the Messiah.
I can relate to Karite Judaism in a way, that they really are seeking the truth in scripture, which I think we should all want. There are Messianic Jewish Congregations that accept Jesus as Messiah, try to keep the Law of God, and welcome both Jew and Non-Jew to learn about Jesus and God's Commandments. Jewish Orthodox, Conservative, and Reform typically won't acknowledge Messianic Judaism as being Jewish, I heard it said, they are like Christians playing dress up.
I really tried to be fair, not to be disrespectful, and present what I believe to be true. I hope it helps answer some of your questions. This is a really deep subject and it's pretty hard to make a blanket statement, add to the fact I never read The Talmud, and never plan to. The New Testament has my favorite Rabbi in it!