J.P.
They do not give discounts on stroller rentals. I would suggest bringing your own or renting one from an outside company in Orlando or Kissimmee since you would have to get inside the park to get one.
You can put your OTC meds in a small cooler (lunchbox size) and leave it in the stroller when you are on rides I am pretty sure that all Disney hotels offer small refrigerators in the rooms with little freezers. All the stroller parking areas have attendants. If you leave the stroller in an area NOT marked stroller parking the employees WILL move it and you will be searching for it.
I have never had my stroller taken or items stolen from it (we have annual passes and visit the parks about once a month), they have cameras EVERYWHERE in the parks and they watch everything that goes on (even on the rides with and without lights).