Yes Maya has radial copy ability. Allow me to demonstrate.
I needed an excuse to create a 202-sided polygon so this is a perfect opportunity.
You have two options:
1. "edit mesh>duplicate special" where you can set the rotation, translation, scale, and number of copies.
2. or use shift-d to copy and remember rotation, scale, and translation. To use this set an objects rotation pivot. Press shift-d to create a copy, then rotate the copy (using the r-key to bring up the rotation gizmo), then just press shift-d again and you will make a new copy rotated the amount of the first copy on each key press.
So first the more important part we need to drill a hole in a surface to create that 202-sided polygon I needed an excuse to create.
Then we add bevels and make the outer surface all wonky to show that the drilled / booleaned surface remains clean and sharp.
As long as the drilled surface is planar, the sice of the value of n in the n-gon does not matter.
This is not to say you'd ever want to create a surface like this one. But in some cases its just impossible to smooth a surface and cut out this kind of shape so if you need to get in close to the edge on a shot you have to put in a lot of edges or you will see corner artifacts and I would prefer to have a final shot that looks perfect then a crappy shot that has geometry that meets some arbitrary rule that dictates the number of allowable sides in a polygon!
This also kinda answered your other question btw. Maya does not have stensil or drill polygon tools. To stensil a hole into a surface you have to boolean, then delete the un-needed faces, then use fill hole.
I agree constrain to background is a very impressive tool. there is a Maya script called shrink wrap but it is cumbersome to use.
"If I have seen further it is by standing on the shoulders of giants." Sir Isaac Newton, 1675