Just checked with a local person. There is NO finished bike path to Mary E. All of the paths pretty well end at Lawson Hill.
Sorry
There's not a paved, or well groomed bike path to Ilium. But there are multiple trails for mountain bikes and hikers that pretty much are a direct shot to town. Pretty certain no vehicles allowed on the trails. We plan to ride to town a few days. I've got all the maps, and stuff.
^ this. Haven't been on them in a while, but pretty sure they exist. If you're well equipped (headlight), relatively skilled or willing to dismount on steeper grades, there's no reason you couldn't take these trails to and fro ... although heading back to Mary E would be easier in terms of exertion ... since you'd be mostly going downhill.
But yeah, no motorized vehicles & you can do what you want re: "golf carts on roads", but I suspect the cops will issue you a ticket if they catch you. This goes for non licensed OHV's too (thankfully, IMO).