Actually I think the Dome of the Rock is exactly where the Temple was as it sits over the foundation stone in which the Covenant of the Ark sat on that rock. There are cut outs that are said to have been made to fit the Arc on the rock.
Plus this is suppose to be where Abraham was going to sacrifice his son at the altar and where David bought the land and built an alter to God to stop the plague. Jacob dreams of the foundation rock where a ladder reached to heaven.
A platform was made to build the Temple and the rock was a central part. I think the rock has to be part of the Temple considering all that history and connection.
Then the Muslims come along 1000s of years later to claim that Mohammad rode his magical horse to the 7th heavens. After taking Jeruslam and see how much religious connection it had to the Jews which they hated. I think this was added to then outdo the Jews as the true religion.
It isn't. This was Herods extension. But its the closest the Jews have to the Temple to pray at. Though as I mentioned it seems there has been a softening for some and they can now go up to the inner terrace where the Dome is. Which would be the Jews most holy place and the inner sancturary of the Temple where the Arc sat.
Yes though I think if you go under the area of the western wall you can see some of the original stones under the stones Herod built from the original walls.
Herod extended the platform but incorporated the original walls surrounding the temple. Some claim that there are original stones and even wood from the Temple.
I watched a doco on when they had to do repairs around the Dome and it showed some original stones and work. They also say that the Octagon building around the Dome of the Rock built by the Caliph Abd al-Malik (688). Was originally a Christian Octagon church as this was a common pattern at that time. The Dome was suppose to be a Mosque but does not have a Kibla.
Its hard to believe that no monument was ever placed there from after the 2nd Temples destruction for 600 years.