Hi Lymecrazed,
First, I am so very sorry you and your family have to go through this. It sure sounds like hell and scary as hell to have your daughter so sick.
For me, while I feel light headed and close to fainting if I get up from sitting or lying down, I have never stopped breathing and passed out like what happened to your daughter. The blue lips are a definite sign she stopped breathing or of her lungs closing up. Could she possibly be allergic to the doxy? When you were on the walk did you notice her having any trouble breathing? Did the paramedics or ER giver her adrenaline/epinephrine? Did they say anything about allergies to medicine?
Again, I am no medical expert, but it sounds like there might be something other than the Lyme in play here. Though I am sure it plays a part as it messes with just about every bodily function.