Don't really have any experience with the SPCA question (I would imagine it depends on the quality of your local SPCA and who they use), but I did want to mention that most of the behavioral issues you are describing need to be addressed with training - neutering (either now or in a few months) will do little if anything to change those as they are not sexually driven.
While it is best to wait until about 18 months (when the growth plates generally close), plenty of dogs are neutered younger than that and do just fine, so while I wouldn't advise doing it in hopes of better behavior, if it is necessary in order to have a place to board him, etc., it wouldn't be the end of the world.