16 gage it looks like. About .060". Sheet metal is low carbon steel, like cold roll. 1010, etc.
No, you don't have to make any corrections. Just make sure the hole is 1/16" bigger than the buttons!
To cut it, you should have special "knock out" punch arrangement. You can drill thru it with 3/8", but you need to go to a store that sells these punches. You drill thru, stick a screw in the hole. Then you put this cutter assembly that makes a knock out hole, like in electrical panels in it.
You won't be able to drill the 1 1/8" hole thru with a drill, it just won't happen.