You need to speak with your sons pediatrician. She will have his history and will be able to help you decide what steps you need to take. She will probably want to see and speak with him and then will suggest some type of therapy.
Just guessing, you are a more mature mom, so you may fall into some patterns of many of us that worry a little too much and make a little too much of events that if we just treated a little less emotionally would help our children.
When the dog died all of you were upset discussing feelings, discussing death, remembering the dog, but at these times, mom and dad need to be "strong" emotionally and not immediately assume that there needs to be an ongoing conversation about the dead dog. A week is a good amount of time for a 3 year old and then drop it from your conversations around your son. If he brings it up, by all means discuss it. Then keep an eye on him to watch his behavior.
At school this is a completely different issue. If you think that he is being singled out for "ALL" of the playground incidents you are not facing the truth. This is not the way daycare works. Day care is a loving caring environment that encourages children to be children. When they misbehave it is a learning moment not a punishment to put him down.
If he cannot respond without overreacting, you need to work on this. You need to correct him and if he overreacts with you, you need to hug him and tell him "HE" is not bad, but his actions are bad. You love him and like when he does nice things. Hitting in school is subject to expulsion in elementary school. In Daycare it is a time out, but tell him to think about why did he want to hit?
Is he frustrated? Why?
Is he mad? Why?
Tell him to "use his words not his hands" to tell his feelings. Give him the words.
Also remind him to keep his hands to himself.
Praise him when he has a good reaction to a correction or a discipline.
Talking about hurting himself, saying "I don't like myself." gets a great reaction from you and dad. For the moment just stay calm and ask him why? If he does not know tell him you love him and dad loves him. Remind him of the good things he does. and move on.
Therapy will also be informative and will give all of you more tools to deal with this. Also sometimes you may have to remind others that just because your child is "big" compared to other kids his age, he is still only 3 1/2.