Can you reserve a stroller and pay for it before you go to WDW?

I don't think so - but I think they have the option that you can pay for your whole visit at once, and then you don't have to wait as long the rest of the time.

Also, you can't take the strollers out of the parks, so if you're going to want it to bring a sleeping/very tired child all the way to the bus stop/car, you'll need to bring your own.

Yes, you can pay for the stroller for your entire trip. You get a "ticket" for each day you are there. You also get a discount when you purchase multiple days. It does make getting the stroller quicker, but you can't take them from park to park. If you leave MK and go to EPCOT, you turn in your stroller at MK and then present your stamped ticket at EPCOT and get a stroller there.