Q1: Is the Ark of the Covenant connected to any unfulfilled prophecies, if not, the ark is not needed.
Besides the ones listed in the OP, Jeremiah 3 and Amos 9.
Jeremiah 3:16 “Then it shall come to pass, when you are multiplied and increased in the land in those days,” says the Lord, “that they will say no more, ‘The ark of the covenant of the Lord.’ It shall not come to mind, nor shall they remember it, nor shall they visit it, nor shall it be made anymore.
The first 20 chapters of Jeremiah are a very severe rebuke to Israel after their falling away toward the end of the tribulation. Many don't understand how eternity works and think God speaks thru His prophets primarily to the people of that day. So I didn't use this in the OP since most assume this is about history.
Toward the end of Jacob's trouble, they will "no more say 'The Ark of the Covenant.'" This implies they had been saying it in the days before this falling away.
Amos 9:1 LXX I saw
Yehovah standing on the altar: and He said, “Strike the
mercy-seat, and the porch shall be shaken: and cut through into the heads of all; and I will slay the remnant of them with the sword: no one of them fleeing shall escape, and no one of them striving to save himself shall be delivered.
The mercy seat sits on top of the Ark, which will be struck in the day of His visitation. Then sudden destruction will come upon them. This is the day that the covenant will be broken, and Jesus will take the Ark to heaven until they repent. That is how the Ark gets to heaven before the 7th Trumpet, by the way.
Most of the modern manuscripts use a different word here from the LXX, so I didn't use this either in the OP.
Q2: If Israel builds a new temple and puts the original ark in the sanctuary, can the Antichrist enter the sanctuary where the ark is. I would think the ark would prevent the Antichrist from fulfilling several prophecies.
Just conjecture on my part.
Good thinking. I partially covered this above. Jesus will come and catch Israel worshipping idols in the day of visitation. (Moses coming down from Horeb and catching Aaron et al worshipping the golden calf and then Moses smashing the 10 commandments/covenant was the sign for this day) Jesus will then break the covenant with Israel and take the Ark to heaven with Him. Then the man of sin destroys them and the Temple. Not one stone will be left upon another. So, the Ark will not be there when the man of sin enters the Holy of Holies.