[prairie]

Planting into corn stalk ground used to be the most common way hay was established in my area, and is still done quite often. Many, including me, have gone to planting into soybean stubble.
It is usually best if the stalks can be shred, then tilled in. If you just till down the stalks it is harder to get a nice seed bed, and you will usually end up with stalk pieces in your first couple of hay harvests.
What herbicides were used on the corn crop?

 

[prairie]

You should be fine with the Roundup.

 

[prairie]

You are correct, only Roundup will be fine.
But many times a preplant or post plant herbicide is put down to control early weeds, followed by 1-2 applications of Roundup.
Most of these preplant and post plant herbicides have excellent residual activity that may carry over into the next year.