Well, I have a three year old, and his responses to things sound pretty normal. How do you respond to them? Children this age do not have a lot of emotional control yet. They have tantrums and patience is definately not a strong point. We can teach them by our reactions to their behavior.
Dont' yell or scream. That sort of reaction tends to make things much worse. Stay calm and in control, and try an immediate consequence (pick your battles, if it were me, I'd start with the biting). If my son does anything to harm another child, I calmly pick him up and say "no, we dont hurt" and he goes to his room. No fighting, no arguing, no getting angry. He comes out when he is ready to apologize and calm down. This is usually 2 to 3 minutes. Tantrums get a calm response to. Something like"I'm sorry, I can't understand you when you scream and yell. When you are ready to use your big boy words and tell me what's wrong, I'll listen". And when he does calm down and talk to me, I make sure I listen to him. It doesn't mean he gets his way, but he does get heard. If I get upset or angry it just escalates the situation. Always stay calm and I am a big believer in age appropriate consequences.
They don't listen well yet. That is normal. So are tantrums and other behavioral issues. I agree, talk to your pedi, but also look at your own responses to his behavior. I know it's hard to keep it under control - we are all human and 3 year olds can be frustrating. I have literally bit my tongue and lip and even had to walk away for a minute or two (knowing he is safe), but it's important. Best of luck. I am told they outgrow this...lol.
Talk to his pediatrician, and see what he or she suggests. There are people the pedi can refer you to that will be able to tell you if this is normal or a serious issue.