I would suggest a strait out approach, that's was they like most of the time. Just go up to here and say "hey baby looking good back there. You want to take this party to a more horizontal position?" that should get you lade whit any asteroid in the universe although I do have to warn you that asteroids from the region Alfa Centauri can be a bit of a pain in the back so stay clear of them.
Carbon-14 dating would be the most appropriate radiometric dating method for dating artifacts found at effigy mounds. This method is commonly used for dating organic materials such as wood, charcoal, or bone, which are typically found in archaeological sites like effigy mounds.
This is mainly because the asteroid belt is thinly distributed (i.e it has a low density). Half of it's total mass is contained within four large asteroids - Ceres, Vesta, Pallas, and Hygiea. As a result, there is only a small chance that a spaceship would be hit by an asteroid; it is estimated that there is less than 1 in a billion chances of such a collision occurring.
Radiocarbon dating.
Then a human would arrive at an asteroid.
It would depend on the size of the asteroid.
Not quite sure what you mean with hang from it; the asteroid would attract the building (much less so than Earth does, but still), so the asteroid would be "downward" for anybody close to the asteroid.In principle there is no reason why you couldn't build something on an asteroid, but that would not "hang... from the asteroid".
The method of uranium-lead dating.
Carbon-14
Depends on the mass of the asteroid
No, if the Earth was destroyed by an asteroid we would not be living today.
The asteroid belt is not a single object. You would have to assess the gravity on each asteroid individually.
The asteroid belt is located between Mars and Jupiter.