[HayZee518]

The bowling ball hole is from the old style vent fan without a grease trap in it. The auto damper is mounted outside. It is probably a 7 inch fan with a built in shroud. You'd pull a chain inside and it would start the fan and open the louver. Nutone still nakes a fan of this type.
Fixing the hole - you'd need remove the formica and the outside louver. You'll have to cut three "sleepers" 2X4s that are there for support (they don't bear any weight) Put one inside on the left, one on the right and one in the center. Fasten with sheetrock screws, then sheetrock and siding outside.