Welcome back, Denise!
Disney Cruise Line does offer a limited amount of complimentary strollers for guest use. These are available on a first come, first served basis at Guest Services. If you think you'll need one and don't plan on bringing your own, be sure to head to Guest Services just as you walk on the ship so you can get one.
But if you have your own, and think you'd like to use it in the airport and in the cruise terminal before boarding and after disembarking, you can absolutely bring yours! Many guests do bring their own strollers, and one of the most popular ones I've seen guests bring are the Maclaren strollers. They are small and foldable like umbrella strollers but also have extra storage that most other umbrella strollers don't offer. These and any other type of compact strollers will fit nicely under the bed. As you mentioned, guests cannot leave strollers in the hallways. Other guests have stored strollers in the closet, in a corner of the stateroom or even gently settled into the bathtub while it's not in use.
The Disney Dream has some wonderful water spaces up on deck. Besides the Mickey Pool, Donald Pool and Mickey slide which are for families and children who are completely potty-trained, there is also
Nemo's Reef splash area. I love that Disney has provided this space for the little ones who still need swim diapers. Since, due to health regulations on cruise ships, children who are not potty-trained and still wearing swim diapers cannot enter the pools. Nemo's Reef gives those little ones a place to cool off and have some wet and wild fun. They won't miss the pool at all once they spot all the fountains, geysers and other water fun waiting for them with Nemo and his friends!
Let me know if I can help with any other questions as you continue to plan, Denise!