What Aladdin saw in Jasmine:
What Jasmine saw in Aladdin:
Well, he did save her hand from being cut off.
How the relationship was strengthened:
With the help of the best musical number, he becomes a prince! But more to the point, he agrees that, yes, she should have freedom to see the world and make her own decisions (not unreasonable mind you). Whereupon she decides maybe she does like him, and wants a ride on his magic carpet (the idea of how he broke in to her bedroom in the first place didn't seem to occur to Princess Jasmine). So, he agreed with her, let her express her feelings, and took her for a ride out of the palace. Things which she seems to have been denied most of her life.
Though he still lied about the prince thing.
The crisis for the third act:
Not so much a relationship crisis, as Aladdin realising he probably should have told the truth and Jafar stealing the lamp to become all powerful.
The Sultan figures, while the law states Jasmine must marry a prince, as an all powerful ruler, he can change that. Which he does.
I can see it working, but they don't really know each other particularly well. They both want aspects of the life the other has (security, freedom), but the amount of time they spent together was really limited to running away from security in the market, a magic carpet ride to Greece and China, and battling a giant snake. They work well together, and seem to get on. And I guess it can be said they both have altruistic morals (such as feeding the orphans, freeing the Genie). Will it last? Unlike other Disney movies, probably.