The simple answer is no. The third molar area you have on photograph most likely represents a mixture or healing tissue, blood clot and possible food debris. Dry socket is ussually with progressive pain that continues to worrsen as several days pass. Having pain while biting hard is normal for recovery.
I would see your restorative dentist as soon as possible. A loose implant can be either the connection between the crown and implant is loose and the crown is pinching your gums or the actual implant may be loose and needs to be removed and redone.
Dental implant recovery depends on the type of procedure you will be having. In general, Dental Implant Recovery and Down Time is minimal. If your procedure is a straight forward implant, you should expect minimal down time and recovery. Most patients can have the procedure done and be back to work the next day. Eating after a dental implant is pretty much returned to normal after a day. If your implant was placed with a healing cap or crown on it the same day, you will most likely be advised not to eat on the implant for several months.
I am sorry to hear about your child. This is something that routinely presents itself in a oral surgeons office. Since the tooth is not in the socket, your child will have bone loss in the future. There is nothing that can be done to prevent this from occurring. You are going to have to wait till she is 18 or stops growing to place a permanent restorative solution. One positive note, medicine is always dynamically changing with new approaches and techniques to restore teeth. 12 years from now, we may not even be talking implant and bone grafting. It could be a whole different approach by the time your child is ready for a solution.
1mm of bone is very minimal bone. Grafting in that area will not take. There are plenty of procedures and treatment plans that can be done. It's the cost that will get you. The question is where is the 1mm bone, how long a span and is there any bone in the posterior. You may be a good canidate for a zygomatic implant as well.
