The strollers in MK are hard plastic and come in single and double models. They have a sun shade above and pouches to hold small purses or cameras in the back. They are fine for toddlers but not for babies. They do not recline.
You pick up the stroller on your way into the park, so you should be able to get one before your CRT breakfast. There is "Stroller Parking" right outside the restaurant in the castle so you can leave your rented or personal stroller there.
If you take the Railroad, you leave your rented stroller where you get on and most times you can pick a new one up at the stop you get off. I have had no trouble at Main Street or Toontown - Frontiertland is more of a challenge. You take your name card with you to put it in your new stroller. If you don't see one, ask a CM nearby and they can help you.
If you have a stroller that folds up, you can take it on the train. Keep in mind that if the train is crowded you might he holding it right in front of you, so only do it if it's small enough folded up.
If you get there at 9:15 a.m. they will certainly have strollers. I have never not been able to get one, even when arriving at a park in the afternoon. But I wouldn't say it's impossible to run out of strollers.
You can also
rent a stroller for the length of your stay. When you do that, you do get a bit of a discount and you can bypass the first part of the line after your first rental.
If you have any other questions, let me know.