God still makes use of the law to convict and draw people to Christ (Gal 3:24) which is its ultimate purpose. But in terms of whether God still honors the old covenant after the cross, the short answer is 'no - God calls all to believe in His Son and come under the new covenant'.
Here is the longer answer... To set a starting point, there are various covenants in what we call the Old Testament. Most have been named by scholars after the person that was involved in the covenant. For example there is the Noahic covenant, the Abrahamic covenant, the Mosaic covenant, the Davidic covenant etc. Some of these covenants are conditional covenants (they relied on certain conditions to be kept by man for the covenant to be honored) and some were unconditional (God promises to do something and it will be fulfilled despite the failure of man). More info on this here if interested: https://www.gotquestions.org/Bible-covenants.html
When you ask 'Is unbelieving Israel still under the old covenant' you are referencing the Mosaic covenant. This is the covenant made with Israel through the giving of the law at Mount Sinai. This was a conditional covenant. And those conditions were not met. In Jeremiah God announced that the day is coming when He would make a new covenant which would not be like the Mosaic covenant because they broke that covenant. This new covenant would be an unconditional covenant based on the promises of God alone.
Jer 31:31-34 The time is coming, declares the LORD, "when I will make a new covenant with the house of Israel and with the house of Judah. (32) It will not be like the covenant I made with their forefathers when I took them by the hand to lead them out of Egypt, because they broke my covenant, though I was a husband to them, " declares the LORD. (33) This is the covenant I will make with the house of Israel after that time, declares the LORD. "I will put my law in their minds and write it on their hearts. I will be their God, and they will be my people. (34) No longer will a man teach his neighbor, or a man his brother, saying, 'Know the LORD,' because they will all know me, from the least of them to the greatest," declares the LORD. "For I will forgive their wickedness and will remember their sins no more."
The new covenant is referenced in various places in the New Testament such as Luke 22:20; 1 Cor 11:25; 2 Cor 3:6; Heb 8:8; 8:13; 9:15; 12:24. Jesus Himself spoke of the new covenant in relation to His sacrifice on the cross:
Luk 22:19-20 And when He had taken some bread and given thanks, He broke it and gave it to them, saying, "This is My body which is given for you; do this in remembrance of Me." (20) And in the same way He took the cup after they had eaten, saying, "This cup which is poured out for you is the new covenant in My blood.
So Jesus' death was the beginning of the new covenant and was ratified by the shedding of His blood. This is a covenant that God wants both unbelieving Jews and unbelieving Gentiles to come under. Now one of the passages in Hebrews speaks directly to your question of whether unbelieving Israel is still under the old covenant. From a human perspective, if they are an observing religious Jew, then they would say they are. But from God's perspective, He now honors the new covenant, not the old. While there are obvious similarities in morality between the two covenants (because God doesn't change) scripture doesn't teach that God now honors two different covenants. The actual fact is that the new covenant replaced the old. With the beginning of the new covenant the old covenant was made OBSOLETE. This is the direct words of scripture:
Heb 8:13 By calling this covenant "new," he has made the first one obsolete; and what is obsolete and aging will soon disappear.
It is interesting that most scholars place Hebrews as being written between 63-65 AD. So when the writer says that the old covenant is obsolete and will soon disappear - well, it was only 5 years later in 70AD that the Romans sacked Jerusalem and destroyed the temple. And the Jews have never had a temple, priesthood or animal sacrifices that the old covenant required, since that day. This is by God's doing. In fact, there is also evidence in their own writings (the Talmud) that God stopped accepting the Jewish sacrifices (which freaked them out!) 40 years before the destruction of the temple (ie from 30AD - around the time that Jesus was crucified). From their own writings:
From the Jerusalem Talmud:
"Forty years before the destruction of the Temple, the western light went out, the crimson thread remained crimson, and the lot for the Lord always came up in the left hand. They would close the gates of the Temple by night and get up in the morning and find them wide open" (Jacob Neusner, The Yerushalmi, p.156-157). [the Temple was destroyed in 70 CE]
From the Babylonian Talmud:
"Our rabbis taught: During the last forty years before the destruction of the Temple the lot ['For the Lord'] did not come up in the right hand; nor did the crimson-colored strap become white; nor did the western most light shine; and the doors of the Hekel [Temple] would open by themselves" (Soncino version, Yoma 39b).
For a discussion of what this means, have a read here (it is interesting): http://www.windowview.org/hmny/pgs/talmuds.30ce.html
While we can learn a lot from the old covenant and the Old Testament in general (especially as it points to Christ) the old Mosaic covenant is obsolete. God is now calling everyone, Jew and Gentile, to believe in His Son and be part of the new covenant.
Act 17:30-31 In the past God overlooked such ignorance, but now he commands all people everywhere to repent. (31) For he has set a day when he will judge the world with justice by the man he has appointed. He has given proof of this to all men by raising him from the dead."
Now as to the future, well, the nation of Israel will come into this new covenant that was prophesied by Jeremiah. And other covenants with the nation of Israel will find their ultimate fulfillment like the Davidic covenant. This is what world events are leading up, and heating up, to! But it is not going to be easy for Israel. It will be the time of Jacob's trouble. But at the end of it, 1/3 of the nation of Israel will be brought through the fire, refined and redeemed! (Zech 13:8-9). These will look in faith to, and then upon, the Lord Jesus, whom they pierced! And the mourning will be great as they realise what they as a people did to their own Messiah... but God will pour out a spirit of grace and cleansing from their sin! They will, as a nation, come under the new covenant which will have it's ultimate fulfillment for Israel in the Messianic Kingdom.
Zec 12:10-13:1 And I will pour out on the house of David and the inhabitants of Jerusalem a spirit of grace and supplication. They will look on me, the one they have pierced, and they will mourn for him as one mourns for an only child, and grieve bitterly for him as one grieves for a firstborn son. (11) On that day the weeping in Jerusalem will be great, like the weeping of Hadad Rimmon in the plain of Megiddo. (12) The land will mourn, each clan by itself, with their wives by themselves: the clan of the house of David and their wives, the clan of the house of Nathan and their wives, (13) the clan of the house of Levi and their wives, the clan of Shimei and their wives, (14) and all the rest of the clans and their wives. (13:1) On that day a fountain will be opened to the house of David and the inhabitants of Jerusalem, to cleanse them from sin and impurity.
I hope that helps.