The "rock" in the Dome of the Rock is the Foundation Stone. Many Jewish scholars believe that this is where the Holy of Holies of the Temple was (like everything in Judaism, others disagree). Many Muslims believe that this stone is the exact point that creation began.

Jerusalem was important to Muslims as well. Before Muslims faced the Kabaa in Mecca for prayer, they (like Jews) faced Jerusalem. They certainly were facing specifically the Temple Mount (the site of the Temple), though I'm unsure whether they faced the future site of the Dome of the Rock, or the future site of Al-Aqsa Mosque (they're separate building on the Temple Mount). I would guess that they actually faced the future site of Al-Aqsa Mosque ("the Furthest Mosque", where Muslims believe that Muhammed a spiritual pilgrimage and ascended to heaven from).

Jews offered no resistance that I'm aware of. The Temple area was likely full of trash during the 7th century, and Muslim tradition records that the Caliph Umar himself lead the clean up soon after the Muslim conquest of Jerusalem. I believe that there had been no formal Jewish place of worship allowed there since the Romans destroyed the Temple in the 2nd century. Once Christians came to power in the Roman/Byzantine Empire, I'm not sure they built any particularly significant buildings there, either. After the Muslim conquest, there were no large Muslims buildings there (though Umar built a small one) for a decades, and the Dome of the Rock were only build at the end of the 7th century and Al Aqsa being built at the beginning of the 8th century, after Muslim rule had been well consolidated in the region (Umar conquered Jerusalem in 638 CE). It is recorded that Umar declined to build a large prayer house there initially so that no one would think that they were praying toward the Foundation Stone rather than the Kaaba in Mecca.

As for what would happen if it were destroyed... In 1949, when Israeli and Arab force reached a cease fire, the Jordanians were in control of the Old City of Jerusalem, which includes the Temple Mount. When Israel conquered East Jerusalem (including the Old City) in 1967, they cleared the whole area in front the Wailing Wall at the Base of the Temple Mount and returned it to a Jewish place of worship, but left the top of the Mount under the control of a Jordanian pious foundation (waqf). The waqf is responsible for maintenance of the Muslim buildings on the Temple Mount (with complex archeological and historical preservation guidelines, I believe set by the Israeli government), so if the two mosques were destroyed, presumably the waqf would rebuild them.

Among (Orthodox) Jews, as above, there's a great amount of debate about everything. The majority opinion is that the Third Temple will be built during the Messianic Era, though an influential minority opinion says that Jews should try to rebuild the Temple when possible. This "when possible" is very complicated. The majority opinion is, because it's not clear exactly where the Holy of Holies was, Jews should not even ascend the Temple Mount because of ritual impurity. The minority opinions outlines where it likely was, and say Jews can walk around most other places. There are groups specifically geared towards getting everything possible ready for the possible rebuilding of the Temple, most notably the Temple Institute. One big issue is finding a red heifer (a purely red furred cow), the ashes of which are necessary for ritual purification. Most Orthodox Jews believe that only the Messiah can lead the rebuilding of the Temple (some others believe that the Third Temple itself will descend from Heaven during the Messianic Era, and the Messiah will just lead the reestablishment of Temple services) so, earthquake or not, the mainline of Orthodoxy would believe building the Temple would be inappropriate without the Messiah (though, again, a well-funded minority would want to try to build it as possible).