I think it is possible he was there as a young boy, but it is a richer story if he was there trying to get Bae. It parallels his current situation trying to get Henry.
I also think the doll could have been Bae’s as it didn’t seem big and could have been tucked inside a shirt. If you knew you had to leave behind all you knew, wouldn’t you take something as a memento that was small and easy to cary? If that doll was the first toy that Bae had, especially one that Rumpel made for him, Rumpel would know what it looked like and meant.
As for the doll being too fragile/squished–here is what it says on Ehow about Straw Dolls:
Dolls made by hand can be treasured by you now and handed down through many generations. One of the easiest types of dolls to make is the straw doll. They take very little time and can be dressed like any other doll. Learn how you can make straw dolls that will put a smile on any child’s face.
If it can be handed down many generations, doesn’t sound that fragile. And yes he was in his dressing gown, but he might have had it in his pockets as his one reminder of home.
“If I had a world of my own, everything would be nonsense. Nothing would be what it is, because everything would be what it isn't. And contrary wise, what is, it wouldn't be. And what it wouldn't be, it would. You see?” -- Lewis Carroll, Alice's Adventures in Wonderland & Through the Looking-Glass