Most of the original questions were answered by a previous poster, but I'll expand on a couple of things. Car seats are not allowed on the regular buses, and strollers must be folded up. Seats are limited on the buses, so if you need to sit while holding the little one and the bus has no remaining seats, you can wait for the next one. We've done that on occasion. On the monorail, strollers can usually remain open with kids sitting in them, but sometimes during busy tines, they will request that they be folded. For the Magical Express between the airport and your hotel, you can have strollers and carseats, but they will be placed underneath the bus. Car seats are not permitted to be used on the Magical Express. If you will be travelling anywhere by taxi, uber, etc, the 1 yo will need a car seat. Sometimes, some of these services will allow you to request a car seat when you reserve the ride, but they aren't always available, and you can really guarantee the cleanliness/condition, etc.
In reference to strollers, I'd recommend bringing the stroller the little one uses at home. With him turning 1 during the trio, I expect that he will either be new to walking or not quite there yet, and navigating through the airport will be much easier with a stroller. It can be gate checked for free, which means you can use it up until you board the plane and it will be there waiting for you when you get off. Plus, using a stroller that you are used to is huge when walking 5-10 miles a day in the parks.
