I've restarted the game and I know have her as my housecarl but she wasn't waiting by the Dragonreach doors, as she had in other games. I finally found her in Breezehome, which I don't own yet. So how can I get her out? I'm nowhere near ready to buy the place.

how did you get into breezehome without the key? which you get when you buy it? you sure you didn't buy it?

Are you the Thane of Whiterun? Do you have the Axe of Whiterun? You get Lydia when you become the Thane. Maybe that's where Lydia is hiding before you become a Thane.