I have developed painful knots on the bottom of both of my feet. What can they be, and is there any treatment?
They could be something called plantar fibromatosis which produces mostly benign tumors in the bottom of the feet. In addition, poor fittling shoes can cause hardened patches of skin called calluses that can cause pain. (Poor foot attire is the number one cause of foot pain.) You'll need to see your GP or a podiatrist to find out what's going on, but in the meantime you might want to check out Health Central's section on Foot Pain...
I do hope you figured out the best path for you, and I hope you will share what worked (and what may not have worked) for you.