I agree with the others to look into medical issues that may be adding to these behaviors. In the meantime:
I see so many kids of daycare providers that do have a hard time sharing their home and mom with other kids. It makes it harder for difficult children.
I don't have a daycare, but my difficult child hates it whenever I invite another kid to join us in any fun - movies, swimming, ect. I love to be able to include several kids in an activity. I thought my kids would like a friend join them but no, they want me to themselves.
I don't have advise for the day care moments. Maybe another daycare mom will have ideas on how to give those special mom moments without the other kids complaining because you favor your own child. The other kids will get resentful over any special priviledges you show your child and the riff will increase. There has to be a balance somewhere so your child feels like your child but not too seperate that the others grow to hate him.
With kids around all week long, difficult child looks forward to having you and the house to himself on evenings and weekends. He may need more alone time with you and at home.
I think most kids do not look others in the eye when apologizing. I do agree that at this age, you are teaching what an apology is and when one is in order, not necessary if they understand how they are suppose to feel. At 3 years old, it will take a lot for him to wish he hadn't done it because he is acting on his feelings and still believes he has the right to do what his body tells him to do when angry or frustrated. I like the idea of drawing a picture. He is not only giving a verbal apology but something material helps both him and the other young person see there is an actual exchange of goodwill.
Keep us informed as how your evaluations go and what help you do find. As you can see, I am not knowledgeable on the medical side but do like to give input on the behavior side. I like to hear how others deal with behaviors because it gives me ideas on how to deal with my children's behaviors.