Can a diabetic ulcer heal by itself?

Yes. As long as there is enough blood flow (circulation) and the wound is protected and taken care of and there is no underlying infection of the surrounding bones, the chances of healing the ulcer are fairly good. Often the blood flow has to be improved, antibiotics are administered and protective footwear is worn.
Sometimes. The ulcer typically occurs due to a minor injury or pressure. Having neuropathy some diabetics can not feel the pain and so the injury continues to happen. Healing is as much related to identification of the ongoing injury as it I making sure that there is circulation in the foot to heal the wounds. This is a very complex problem and requires involvement of multiple specialists to achieve success.
Wound care.. Diabetic ulcers will generally heal if you offload the area (decrease direct pressure with an insert, or a wheelchair), have your doctor trim the callus and dead tissue away on a regular basis, and if the ulcer is infected, you may need oral or IV antibiotics, depending on the severity of the infection.
No. See podiatrist. Control blood sugar. Don't treat by yourself. You are at risk for amputation.
Maybe... Diabetic ulcers are generally caused by damage to the small blood vessels of the foot. Poor circulation prevents even minor trauma from healing-even just pressure- and promotes infection. If you are a smoker, stopping this can help greatly along with a strong wound care regimen and quality footwear. Not reversing the causes, though, it is unlikely to heal well.