My son is 3.5 yrs old and hurts the kitten i am wandering is something wrong with him he still has tantrums?

My son is starting to concern me he hurts the kitten throws the biggest tanties.He cries to get what he wants,he hits me and threatens his dad to hit him but never does.I have smacked him THIS DOES NOT WORK i am not sure what else to do.I have spoken to a behavioural therapist and she suggested monitor his diet and his sleeping pattern.I have done this and i find his behaviour is far more worse when he has had less than 9 hrs sleep.He also expects something when we go out to the shops otherwise throws a tanty uses foul language and just recently starting to hurt the chn at d/care but when i ask the carers why they cant tell me waht occurred only that they caught him (so i dont know what the other chn are doing to him cos he seems to get caught)He is infatuated with spiderman and pretends to talk to him(I know this is age app as he is role playing).I took him to the bank the other day and the teller made her own diagnosis saying he had aspergers cos he wouldnt sit still