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When it comes to wild animals, my recommendation is to either fix them and release them or euthanize. I don't think you're giving him a good quality of life just living in a cage or box. It's different with captive raised birds. When they are wild birds with injuries, you really either have to repair and release or do the humane thing and euthanize.
The leg can probably just be amputated. The wing needs x-rays to determine if there are fractures. If there are wing fractures, they can be pinned and then the pins removed in a few weeks when things are back to normal. At that time, he can be released.
I hope this helps, let me know if you have any other questions.
I would just pick up a pelleted bird food that is complete and balanced to feed this pigeon. That would be better than a fledgling formula.
I don't think I would start antibiotics at this point. I don't usually do that until the actual infection is taken care of. Otherwise, we may be breeding resistant bacteria.