yes and yes
They will check your bags as you enter the park. As long as you aren't bringing a cooler full of ice and enough pic nic supplies to feed an army, they won't say anything to you at all. They are very understand of bringing snacks and drinks for little ones.
There are many benefits to bringing your own stroller. Cheaper. They are more comfortable, (rentals are hard plastic) you can take it to the bus or parking lot, then to your hotel room.
You must take your child out and fold it up while on the buses. You can usually roll it right onto the monorail.
Bring a rain cover for your stroller...and cover it when you go into an attraction. It can be sunny one second, and raining the next. Soggy strollers are no fun.