Rose stroking the TARDIS was a wondeful moment. Iloved it.

yeah, I thought it was her name aswell,but Bad Wolf is so much better!
and when Rose started with the technobabble,that made her the Doctor for me.

well,the Doctor and Rose never really defined the relationship. I don't think we as human have the right word for what they are,anyway.so it would've taken months just to explain the history,let alone answer the question!