Since you have tried many things and they don't seem to be helping here are a couple of new ideas. When toddlers begin exploring they try different behaviors to see what the reactions will be. Your son has found that if he does certain behaviors he gets a reaction. It most likely isn't that he wants to hurt someone as he wants to get the reaction. The first thing to do is to stop reacting as if he has done something bad. He will pick up that the message here is that he is a bad boy, even if you don't use those words. Once he buys into your "belief" that he is a bad boy he will continue along that line just so his actions now match the belief that he has taken on. Don't worry it is not too late to change this. Just go into the situation without any judgment. The way you respond will speak louder than any of the words you say. When he bites someone, simply remove him from the other person and offer a different activity away from others. Not as a punishment. Your tone will make the difference.Calmly let him know that biting, hitting, etc, are not allowed. Ask him what else he can do that would be fun. If he seems to be angry about something see if you can find out what he might be upset about and just doesn't have a voice for telling you. Kids at this age use their behaviors to communicate when words don't work. He needs to feel safe and secure in how he feels and when we punish kids without getting to the reason for their actions, they get a message that it is wrong for them to feel. You might also watch and see how the other adults at the daycare respond to him. He is taking in their reactions too.
As for when he screams when you tell him "No" then find another way to tell him no. Let him know that you aren't telling him no, you just can't do what he wants at that time. Using responses like - not now, maybe later. that isn't safe, what else can you play with. that doesn't work for me, or sometimes depending on the situation, offering another choice- it's too cold for the park today, would you like to make a fort under the table?
Finally don't be too hard on yourself. Asking questions is always good- there are no bad questions.