What is the proper way to cut a hole in a window pane?
Assuming not tempered, this is relatively easy, if tedious. If tempered, you'll have to replace the pane, as it will shatter as soon as you try to drill a starting hole. Would be easier out of the frame, though (or you could simply have a glass shop do for very easy, but probably more cost).

Diamonds are not needed to cut glass.

Drill a hole large enough to feed a carbide rod saw through. With care you can use one without a hacksaw frame, and you'd pretty much have to do that if not removing the glass from the frame. You can cut a lot faster if it's held in a hacksaw frame. Pull, rather than push when cutting, and be aware that a rod saw will cut in any direction, which is good for making your rounded corners, but requires attention when cutting straight lines. Used without a hacksaw frame you will also need to be careful not to bend it too much.



I have done this a few times on custom fish tank covers to make cutouts for filters.