35:00 about building. Hatam Sofer answers that Kohanim can and should do it. If not available then Leviim, if not them then Yisraeli can do it.
About the Temple Mt. All we need to do is to do some digging and see what is found and then make measurements from their. The Machloket is where exactly is the spot located. Some believe its under the dome and others believe its in between the 2 mosques (strong sources available).
Takes just a little digging and once its done it wouldn't be too hard to measure and know from there.