If an MRI shows infection in the bone of a foot of a diabetic, can they recover without losing their foot?

Diabetic foot. You do not give enough information to give a best answer. There are multiple bones in the human foot -- 26 to be exact! it depends on where the bone infection is, how long the infection has been in the bone, and whether there is an ulceration and or cellulitis associated with the bone infection. Many times there may be loss of the bone involved without the loss of the foot.
Maybe. It depends on the severity of the infection and arterial blood supply to the affected foot. Some infections will respond to antibiotics and some infections need to be treated with surgery, debribement and possible amputation of infected bone.
IV antibiotics. If your doctor determines that the bone infection could be cured by antibiotics, then you will need 6-8 weeks of IV antibiotics. However, you need adequate circulation for IV to work. Hence, a team approach is needed. Many docs need to get involved, including podiatrist, vascular, primary care and infection disease.
Yes. This type of patient needs coordinated care from a podiatrist and an infectious disease specialist and possibly a vascular surgeon. They usually need prolonged intravenous and oral antibiotics and surgical debridement and wound care. Often this treatment is curative. Some times toe, ray, transmetatarsal or even a below knee amputation is necessary.
It is possible. If appropriately treated it may be possible to save the foot. Have it evaluated by a professional and do not delay treatment.
Possible. Sounds like you have osteomyelitis, bone infection. If the infection is mild enough, it can be cleared with antibiotics. In more severe cases, antibiotics don't work, and amputations are a consideration.
Yes. If shoes with a soft upper or extra depth toe box still irritate your deformed toes and you are healthy enough, there are surgeries that can correct the deformities. Stretching exercises rarely correct the deformity or toe appliances. At best they prevent progression of the deformity. The type of surgery depends on the type of straightening that is needed.
Yes. There is treatment for osteomyelitis of the foot. It often calls for surgery to cut out the infected and dead bone(debridement) followed by the appropriate antibiotics for 6 weeks or more. The success rate varies from 30-85%. How healthy or conversely how sick the patient is also impacts on the decision. If the infection is life threatening amputation maybe best.