Is it right to ask a co-worker if they consider you a friend or acquaintance?

[Willow]

If you have to ask, then you are an acquaintance. Real friends don't ask that question of each other, they just know.
You have not become "closer", simply more at ease with one another, you have no contact outside of work other than a social network site. Real friends meet up outside both of these and share family and other friends with each other.
I think you know where you stand at the moment with this person, I also think you would like to actually be her friend.
So get more acquainted, share more of yourself with her, offer to meet up outside of work.
If you get a positive response then she may well become a friend, a negative response will tell you that she is happy just being a work colleague who is also an acquaintance.
It may well be that those "hidden" pics are for family members only.