What is the smallest possible cup size for a breast reduction, given my size and type of breasts? (Photo)

Breast reduction eligibility and coverage by insurance varies from one insurance carrier to the next, and cannot be determined based on cup size, but only based on an in-person examination to figure out how much tissue the surgeon needs to remove.  You also often have to qualify for multiple other criteria from your insurance carrier, so be sure to request their breast reduction coverage policy ahead of time before your plastic surgery visit to ensure you meet all the other non-size related demands that they have.  Be sure to discuss your goals and concerns with your board certified plastic surgeon and consider the pros and the cons. Be certain your surgeon is properly certified.  I've attached a link where you can start your search.  

Thank you for the question and congratulations on your decision to proceed with breast reduction surgery. You may be surprised to know that your goals/requests are not that unusual.   I see several patients every year who wish to achieve as small of a breast appearance outcome, for a variety of personal reasons.
 Generally speaking, it is possible to reduce the breasts size very significantly.  Based on the sophistication of your question, I think you may already understand that the concern with the amount of tissue removed is related to blood flow to the remaining tissue; if too much tissue is removed in one operation the blood flow to the remaining tissue (including nipple/areola) may be compromised. Part of the tissue that is left in place is called the “pedicle"; this segment of tissue is responsible for delivering the blood supply to the nipple/areola tissue. If the pedicle is made too small (in the effort to reduce the breasts as much as possible) then patient will likely have problems with tissue survival.   Occasionally, a staged procedure is helpful; in others, nipple/areola grafting is helpful.
I think that you are wise in understanding the importance of preoperative communication when it comes to achieving an outcome that you will be pleased with. In my opinion, successful outcomes with  etc. surgery depend on:
1.  Careful selection of plastic surgeon (and from the surgeon's  standpoint, careful selection of patient).  I would suggest starting with the American Society of Plastic Surgery and/or the Aesthetic Society of Plastic Surgery to obtain a list of well experienced board-certified plastic surgeons. Then, I would suggest you visit a few surgeons whose practices concentrate on aesthetic surgery. ***Ask to see lots of examples of their work and preferably speak/see patients who have had similar procedures done.
2.  Careful communication of goals ( which I will discuss further below).
3.  Skillful execution of procedure ( preoperative, intraopererative, and postoperative patient management).
In my practice, I would ask that you NOT communicate your goals, or evaluate the outcome of the procedure performed, based on cup sizes.  There is simply too much variability when it comes to bra sizes between bra manufacturers and even store employees doing the bra fitting measurements.
 It will be very important to communicate your size goals with your surgeon. Most patients wish to achieve a enough of a reduction to help with their symptoms while remaining proportionate with the remainder of their torso. Again, I suggest that you do not communicate your goals in terms of achieving a specific cup size. For example, a “A or B cup” may mean different things to different people and therefore may be a source of miscommunication.
In my practice, I ask patients to communicate their goals with the help of  computer imaging and/or goal photographs,  as you have done here.  Obviously, the outcome associated with the breast surgery will not necessarily match that of goal photographs perfectly, but they do serve as a better communication tool that subjective terms such as "natural", "essentially no", "flat"… Evaluating goal photographs also allows for a plastic surgeon to determine the consistency of the patient's goals and allows for a discussion of realistic expectations as well.  
***Needless to say, when it comes to achieving patient satisfaction with the outcome of surgery,  it is very important that a patient has consistent goals (fully decided on what she would like to achieve) and a good understanding of realistic expectations (what outcomes can and cannot be achieved). 
 Best wishes with your decision making and for an outcome that you will be very pleased with.