Are the Jews still God's chosen people?

Yes, God chose Abraham and established through him an everlasting nation with the promise of an everlasting covenant (Genesis 17:1-8, 1 Chronicles 16:15-17, Luke 1:54-55). Since then, God has been identified as the God of Abraham and his offspring (Exodus 3:6, Acts 3:13), and as the God of Israel (Exodus 5:1, 1 Kings 1:30, Psalm 72:18, Jeremiah 51:33, Matthew 15:31). Throughout Jewish history, the Israelites continually broke God's commandments and were punished severely, however, God always forgave them and re-established their nation when they repented (Isaiah 30) -- which is the meaning of Israel itself ("he struggles with God" - Genesis 32:28, 35:9-10). This is the name the Lord gave to Jacob, because he struggled with God and men and overcame (Genesis 32:28).

        "Therefore say to the house of Israel, 'This is what the Sovereign LORD says: It is not for your sake, O house of Israel, that I am going to do these things, but for the sake of my holy name, which you have profaned among the nations where you have gone. I will show the holiness of my great name, which has been profaned among the nations, the name you have profaned among them. Then the nations will know that I am the LORD, declares the Soverign LORD, when I show myself holy through you before their eyes.' " (Ezekiel 36:22-23)

Even when the Israelites were disobedient to God, they still remained his chosen people for the sake of the Lord's name (1 Samuel 12:22, Psalm 106:8, Isaiah 43:25, 48:9-11, Jeremiah 14:20-21, Ezekiel 20:44, 36:22-23). After nearly 1,900 years, Israel has been established once again as God has promised -- even in the midst of all its enemies and with a lack of belief of the majority of its people.

        "I ask then: Did God reject his people? By no means! I am an Israelite myself, a descendant of Abraham, from the tribe of Benjamin. God did not reject his people, whom he foreknew." (Romans 11:1-2)
        "Again I ask: Did they stumble so as to fall beyond recovery? Not at all! Rather, because of their transgression, salvation has come to the Gentiles to make Israel envious. But if their transgression means riches for the world, and their loss means riches for the Gentiles, how much greater riches will their fullness bring!" (Romans 11:11-12)

Although John the Baptist proclaimed that God could raise up sons for Abraham from stones (Matthew 3:9, Luke 3:8) and the Apostle Paul plainly states that those who believe by faith are sons of Abraham (Romans 4:16-17, 9:6-8, Galations 3:6-9), this in no way means that God now refers to Israel as the body of Christ and no longer as the Jewish nation he originally established (Luke 2:30-32, Acts 13:26). On the contrary, they still remain the firstfruits of God's chosen people (Jews and Gentiles) and the root of Christianity (Romans 11:16).

        "If some of the branches have been broken off, and you, though a wild olive shoot, have been grafted in among the others and now share in the nourishing sap from the olive root, do not boast over those branches. If you do, consider this: You do not support the root, but the root supports you... After all, if you were cut out of an olive tree that is wild by nature, and contrary to nature were grafted into a cultivated olive tree, how much more readily will these, the natural branches, be grafted into their own olive tree!" (Romans 11:17-24)

Remember that salvation comes through the Jews (John 4:22) and for the Jews first (Romans1:16). Jesus was originally sent to save the Jewish Israelites only (Matthew 10:6, 15:24, John 12:12-16), and it wasn't until after they crucified him that salvation was made available to the Gentiles (Acts 11:1-18). Even so, this was the Messiah's first appearance, and at his second coming he promises to give his seal of redemption with the name of the Lamb and the Father to 144,000 pure members of the twelve original tribes of Israel --12,000 in each literal tribe (Revelation 7:3-8, 14:1-5)-- and to establish a new Holy City of Jerusalem with twelve gates named after the twelve tribes of Israel (Revelation 21:10-12). This means that those Jews who still live by the original covenant will be redeemed and retain their original standing as God's chosen nation.

        "I do not want you to be ignorant of this mystery, brothers, so that you may not be conceited: Israel has experienced a hardening in part until the full number of the Gentiles has come in. And so all Israel will be saved, as it is written: 'The deliverer will come from Zion; he will turn godlessness away from Jacob. And this is my covenant with them when I take away their sins.' [Isaiah 59:20-21] As far as the gospel is concerned, they are enemies on your account; but as far as election is concerned, they are loved on account of the patriarchs, for God's gifts and his call are irrevocable. Just as you who were at one time disobedient to God have now received mercy as a result of their disobedience, so they too have now become disobedient in order that they too may now receive mercy as a result of God's mercy to you. For God has bound all men over to disobedience so that he may have mercy on them all." (Romans 11:25-32)