A Cornish hen, or several, can be prepared on a rotisserie depending upon the size of your rotisserie. You will need to adjust your cooking time to accomodate the size.
I would think you can. The chickens you buy at the supermarket are also pretty small. And rotisserie things come in different sizes. Some toaster ovens even have them.
You should thaw the hen. Apply salt, pepper, and any other seasonings. Place the rotisserie forks through the hen. Check the temperature. The desired temperature should be above 165.