Smithsonian has a really good article on the Temple Mount, it's importance to all 3 religions, what is known, and not known and the conflicts surrounding it.
History, Travel, Arts, Science, People, Places | Smithsonian
The odds and ends I was gathering are the fruits of one of Israel’s most intriguing archaeological undertakings: a grain-by-grain analysis of debris trucked out of the Temple Mount, the magnificent edifice that has served the faithful as a symbol of God’s glory for 3,000 years and remains the crossroads of the three great monotheistic religions.
Jewish tradition holds that it is the site where God gathered the dust to create Adam and where Abraham nearly sacrificed his son Isaac to prove his faith. King Solomon, according to the Bible, built the First Temple of the Jews on this mountaintop circa 1000 B.C., only to have it torn down 400 years later by troops commanded by the Babylonian king Nebuchadnezzar, who sent many Jews into exile. In the first century B.C., Herod expanded and refurbished a Second Temple built by Jews who had returned after their banishment. It is here that, according to the Gospel of John, Jesus Christ lashed out against the money changers (and was later crucified a few hundred yards away). The Roman general Titus exacted revenge against Jewish rebels, sacking and burning the Temple in A.D. 70.
Among Muslims, the Temple Mount is called Haram al-Sharif (the Noble Sanctuary). They believe it was here that the Prophet Muhammad ascended to the “Divine Presence” on the back of a winged horse—the Miraculous Night Journey, commemorated by one of Islam’s architectural triumphs, the Dome of the Rock shrine. A territorial prize occupied or conquered by a long succession of peoples—including Jebusites, Israelites, Babylonians, Greeks, Persians, Romans, Byzantines, early Muslims, Crusaders, Mamluks, Ottomans and the British—the Temple Mount has seen more momentous historical events than perhaps any other 35 acres in the world. Nonetheless, archaeologists have had little opportunity to search for physical evidence to sort legend from reality. For one thing, the site remains a place of active worship. The authority that controls the compound, an Islamic council called the Waqf, has long forbidden archaeological excavations, which it views as desecration. Except for some clandestine surveys of caves, cisterns and tunnels undertaken by European adventurers in the late 19th century—and some minor archaeological work conducted by the British from 1938 to 1942, when the Al-Aqsa Mosque was undergoing renovation—the layers of history beneath the Temple Mount have remained tantalizingly out of reach.
Of note, it does appear that the Palestinians are trying to rewrite the history and I agree that is wrong and I think it is part of each side attempting to hold on to Jerusalem:
Until recently, Palestinians generally acknowledged that the Beit Hamikdash existed. A 1929 publication, A Brief Guide to the Haram al-Sharif, written by Waqf historian Aref al Aref, declares that the Mount’s “identity with the site of Solomon’s temple is beyond dispute. This too is the spot, according to universal belief, on which David built there an altar unto the Lord, and offered burnt and peace offerings.” But in recent decades, amid the intensifying quarrel over the sovereignty of East Jerusalem, a growing number of Palestinian officials and academics have voiced doubts. “I will not allow it to be written of me that I have...confirmed the existence of the so-called Temple beneath the Mount,” Palestinian leader Yasir Arafat told President Bill Clinton at the Camp David peace talks in 2000. Arafat suggested the site of the Temple Mount might have been in the West Bank town of Nablus, known as Shechem in ancient times.
...And in November 2010, the Palestinian Authority created a diplomatic kerfuffle when it published a study declaring the Western Wall was not a Jewish holy site at all, but part of the Al-Aqsa Mosque. The study contended, “This wall was never part of the so-called Temple Mount, but Muslim tolerance allowed the Jews to stand in front of it and weep over its destruction,” which the U.S. State Department called “factually incorrect, insensitive and highly provocative.”
There is a lot not known, because archealogical study of it is so tricky.