If I have a breast reduction, will my breast still be painful and sore and fluxuate in size during menstruation?

Thank you for your question. 
You definitely meet many criteria for breast reduction surgery; painful heavy breast, neck and back pain, and shoulder grooves from your bra strep and would likely benefit from breast reduction surgery. Most of your symptoms (especially pain) would likely significantly improve. The pain with menstruation and the fluctuation in size will likely still be present to some extend after breast reduction surgery. But women in my experience have noticed it to be much less, to a point where it doesn't hold them up anymore from performing their daily activities. 
Of course every woman, including you, is unique. So it is impossible to predict how much relief of discomfort during your menstrual cycle you will have after breast reduction surgery, but almost all the women experience a significant improvement of their overall symptoms (especially pain) after breast reduction surgery. They are able to find a bra in a regular store, are able to exercise and move around easier, the neck and back pain is much better, and the grooves in the shoulder s are resolving. 
I recommend you being seen and evaluated by a board certified Plastic Surgeon, to determine if you would be a good candidate for surgery. He or she can help and guide you through the process, explain the procedure to you, and based on all the information you can then decide if you would feel comfortable proceeding with breast reduction surgery. 
Best wishes, 
Franziska Huettner

Breast reduction surgery can be life changing for many women.  It typically makes an immediate improvement in back, neck, and shoulder pain associated with large breasts and in many women results in complete resolution of these symptoms.  Many patients find it easier to breathe with the weight of their breasts reduced.  With regard to breast pain, however, the results of breast reduction surgery are less satisfying.  Many women with painful breasts, particularly pain that fluctuates with their menstrual cycle, have fibrocystic changes in their breasts that cause the pain.  This is a benign type of breast tissue consisting of scarring and cysts and it is quite common.  Because the breast reduction surgery makes breasts smaller but doesn't change the composition of the breast tissue, it doesn't usually make that pain significantly better.I advise my patients considering breast reduction surgery that if the only reason they're considering the surgery is to improve breast pain, that they probably shouldn't have it done.  If they have other reasons for considering breast reduction surgery (e.g. back/neck/shoulder pain, rashes under the breasts, cosmetic concerns, improving their proportions) and also have breast pain they wish were improved, they can have the surgery and hope that some improvement may be made in the breast pain.

Your remaining breast tissue will likely fluctuate as it did prior to a breast reduction. You will still have breast tissue that is affected by hormones; you will just have less of it. However, with less breast tissue present, you may find that you have less swelling and discomfort around your menstrual cycle. The neck, back and shoulder strap grooving should improve with your breast reduction as well. 

It is difficult to say but most patients with symptomatic problems obtain a reduction in these symptoms following a breast reduction.