Yes. This is usually done with small incremental challenges of the food rather than shots.
No. There are no protocols to desensitize someone to foods via injections. There are oral desensitization protocols, but at present they are only suitable for research studies.
No. Thus far there is no effective means for desensitization to foods across the boards and any attempts at desensitazation should not be tried at home. There are clinical trials in progress some of which are very positive however adverse reactions including death have occurred in the past and therefore until standard protocols are in place this should not be considered. Avoidance only.
NO! Allergy shots for foods is not indicated. Several studies in the past have demonstrated that food allergy shots are potentially very dangerous. Oral desensitization studies are ongoing and look very promising but the data is not yet clear as to the proper dosing, duration and whether this is "curative" or offer some degree of protection for accidental exposures. Google cofars for a good resource.