You definately need your own stroller for you baby because of everything you mentioned. My rule of thumb is if the child is under 2 you probably want a regular stroller. The rentals are hard -- no padding.If there are two adults, you could rent a single stroller as needed for your 4yo.
Have you looked at Maclaren strollers? they are umbrella style, some have padding and seat adjustments...much more expensive than the average stroller. I bought one for my third and I LOVE IT. It is compact because of the umbrella style and holds a child up to 55 pounds. This means you'll be able to use it until your 16mo old old enough to want to walk all the time. (My 7 year old can ride in it if she wants.) You'll get a lot of long-term use out of it. Well worth the price.
Oh I didn't mention steering -- one handed. I have a techno classic -- adjustable handles make it great for my DH (over 6 foot) to push. I like it because it has the padded, adjustable seat. DS can nap in it comfortably.
I bought mine online. There have been other threads on strollers... which list some other sites. I got mine at albeebaby.com. Previous year's model at greatly reduced prices. They have a specialty baby store in NY city. Customer service was excellent. Other models are even more compact. At one point they had a quest at Target at a good price.
Only downside to this stroller is there is no cupholder. We have a skip-hop duo stroller bag that has a pocket that can hold a drink...but I do miss a cupholder.
You definately have several choices...including renting a double stroller from an orlando company that will deliver to your hotel...that would save you a lot of room in the car!!
You've got plenty of time to sort it all out. I'm sure you'll enjoy your vacation no matter what you decide.
