Well, most asteroids are agglomerations - loose rock and dust held together by gravity. Unless the aliens are supporting every part of it, on the inside as well, it’s going to rain gravel on New York. If it’s solid enough and low enough, maybe it would be worth mining?

Gravity would be slightly weaker under it, because of its own gravity, but I think the difference would be small enough that you’d need measuring equipment to see it. No moon hopping.

Weather would be different. I think there’d be more rain on the windward side, because some water-laden air would have to drop its load in order to rise over it. You’d get wind vortices on the other side of it.