Why are the Jews called God's chosen people? Since all mankind descends from Adam (and later Noah), why do Jews get special designation among all other peoples, especially since they worshipped false gods and disobeyed God's word?
First, it's important to distinguish between the Biblical term "Israel" and the modern political state of Israel. The existence of the modern political state of Israel is important since the nation's return to the land in the early 20th Century was a fulfillment of prophecy and a stepping stone toward Christ's eventual return. Nevertheless, the promises of Scripture were spoken to a nation of people called Israel, not necessarily to a specific political entity or even to every Jew. As Paul says in Romans:
Rom. 9:6 But it is not as though the word of God has failed. For they are not all Israel who are descended from Israel;
Rom. 9:7 nor are they all children because they are Abraham’s descendants,
Therefore, in all cases when VBVMI uses the term "Israel," we are referring to the worldwide community of Jewish descendants wherever they may live, not to the modern political state (unless otherwise noted).
The Bible declares that the descendants of Abraham, Isaac and Jacob are God's chosen people, a possession for Himself called to be the "wife" of Jehovah and to bring forth the word of God, the temple service, the covenants and ultimately the Messiah.
God formed this new nation out of nothing by His own choosing so that we would understand that no nation was worthy of God's relationship. God created the nation of Israel, which didn't exist prior to Abraham's line, and He did so to prevent any claims of honor. Had God selected an existing nation somewhere in the earth to bring the Messiah, then that people could claim that they were chosen because they were better than other people.
Nevertheless, the Jewish people are not inherently better people than the rest of mankind, nor did Israel earn (much less deserve) their special status with God. In fact, God declared that Israel was a nation of unremarkable people:
Deut. 7:6 “For you are a holy people to the LORD your God; the LORD your God has chosen you to be a people for His own possession out of all the peoples who are on the face of the earth.
Deut. 7:7 “The LORD did not set His love on you nor choose you because you were more in number than any of the peoples, for you were the fewest of all peoples,
Deut. 7:8 but because the LORD loved you and kept the oath which He swore to your forefathers, the LORD brought you out by a mighty hand and redeemed you from the house of slavery, from the hand of Pharaoh king of Egypt.
The Bible says the Lord selected Israel to be His chosen people despite their lowly and unremarkable status. It was the Lord Himself Who created the Jewish people by way of a promise He delivered to Abraham, Isaac and Jacob. The Lord promised to grant these men innumerable descendants, who would inherit a special place in God's Kingdom.
The Lord's decision to award Israel a special place in His plan was simply an act of grace, not one based on Israel's merit. In other words, Israel wasn't chosen because they were special; Israel is called special because God chose them.
The Bible also says the nation has received great honor and significant advantages as a result of being chosen by God:
Rom. 3:1 Then what advantage has the Jew? Or what is the benefit of circumcision?
Rom. 3:2 Great in every respect. First of all, that they were entrusted with the oracles of God.
Rom. 9:4 who are Israelites, to whom belongs the adoption as sons, and the glory and the covenants and the giving of the Law and the temple service and the promises,
Rom. 9:5 whose are the fathers, and from whom is the Christ according to the flesh, who is over all, God blessed forever. Amen.
Among other advantages, Israel was given the word of God (to guard and deliver to the world), the covenants of promise, the Law and the tabernacle service. Most importantly, Israel was chosen to bring forth the promised Messiah. The Lord elected Israel for the privilege of receiving these blessings, and by His choosing, the Lord made Israel special and set them above all other nations on Earth.
The Lord did this out of love for Israel and out of a desire to use Israel to glorify Himself and bless the entire world. As the Lord declared to Abraham:
Gen. 12:3 And I will bless those who bless you,
And the one who curses you I will curse.
And in you all the families of the earth will be blessed.”
Israel didn't deserve the Lord's mercy (and neither do we), but still, the Lord chose to make Israel special according to His word. He did this despite their history of sin. As you mentioned, the people of Israel have been (and continue to be) disobedient to God and in violation of His word. Still, the Lord remains faithful to Israel, as Paul says:
2Tim. 2:13 If we are faithless, He remains faithful, for He cannot deny Himself.
The Bible teaches that the Lord responded to Israel's disobedience with a temporary judgment. The Lord set Israel aside for a time, while extending His grace to Gentiles in their place. Paul teaches in Romans 11 that this "switch" is only temporary in God's plan. Eventually, the Lord's attention will return to His chosen people, and in that future day He will extend His grace to the Jewish nation again. As Paul says in Romans 11, in a day to come day all Israel (i.e., those who are Israel on that future day) will be saved, because the gifts and calling of God are irrevocable:
Rom. 11:25 For I do not want you, brethren, to be uninformed of this mystery — so that you will not be wise in your own estimation — that a partial hardening has happened to Israel until the fullness of the Gentiles has come in;
Rom. 11:26 and so all Israel will be saved; just as it is written,
“THE DELIVERER WILL COME FROM ZION,
HE WILL REMOVE UNGODLINESS FROM JACOB.”
Rom. 11:27 “THIS IS MY COVENANT WITH THEM,
WHEN I TAKE AWAY THEIR SINS.”
Rom. 11:28 From the standpoint of the gospel they are enemies for your sake, but from the standpoint of God’s choice they are beloved for the sake of the fathers;
Rom. 11:29 for the gifts and the calling of God are irrevocable.
You can learn more about the importance of Israel in God's plan by listening to our Romans study online.