I feed hay inside in Nibble Nets. They are easy to fill & you can hang them anywhere. They are tough too & keeps the horses busy.
A side opening on a run in will prevent a horse from getting trapped if being chased off by another horse entering.
If I had your set up I would hang bags on opposite sides on the inside of your building.
For mud control Ground Master makes a permeable floor which works very well when buried under a few inches of traffic bond or other small crushed stone. There are other permeable membranes that work well too for mud control.
How exciting for you!