You can always check-in early -- as early as 7:30am I believe -- at any resort and at any time of year. The question is whether or not your room will be ready.
When you arrive, you can go to the desk and check in. You'll get your Key to the World card, which will become your room key, so you'll have access to transportation, your park tickets (if you've bought them as part of a package), and your dining credits (if you're doing the
DDP). The person at the desk will tell you if you can go to your room early or, if it is not ready, will give you a phone number to call later in the day to get your room number. When you go back to the hotel later, your KTTW card will open your door so you don't need to go back to the front desk. It's really very easy to manage, so no worries! And since your bags will likely not arrive for a while, there's very little advantage to getting in to the room, I'd think.
That said, there's always a chance the room will be ready. It doesn't necessarily follow that the room will be ready if it's a less crowded time of year or not ready at a more high volume time at the hotels. A lot of factors are involved -- really no way to know for sure.
I don't know about the stroller, but I imagine it wouldn't be a problem. A good rule of thumb seems to be to carry anything you wouldn't want to be without on ME with you. If you want to have access to a bag, it's still better to carry it on ME and leave it with bell services, as you'd be able to get to it if necessary.