Please Help! Does my 3 year old have SPD?

I apologize in advance if this becomes a novel.

My son early on was diagnosed with a communication delay due to having a speech delay. At first I was convinced that after having tubes put in his ears, seeing a counselor and taking speech classes everything else would sort of "fall into place". However, that has not been the case at all.

Here are some melt downs we have had to give some insight to what we are dealing with:

Everynight is a nightmare between dinner and bath time. He will gag himself with his food so that he won't have to eat it (if it is something that is "nasty"). And then at bathtime will completely go into Total Hysterics when it is time to was his hair and face! (getting hair cuts are the same way!)

He also will completely melt down if his toothpaste is not perfectly put on his toothbrush. I mean completely melt down.
Also recently he has (for no apparent reason) picked a whole about a foot long in his door by scratching at the paint and peeling it off. And picked and peeled all the paint out the bathtub. He tares everything and anything up napkins/paper/etc.

I can not count the times we have left a full basket of groceries in the middle of the grocery store because of his melt downs.
He panics if dad/uncle or anyone he's even use to want to play or turn him over or "rough house" with him buy will randomly go up to anyone and head butt them or try to play.

He also recently fell down and skinned up his knees. He kept on going and didn't even notice he was bleeding and had hurt himself. But, if another child even bumps into him he immediately starts crying and says they "hurt him".

That's all I can think of right now. Any insight anyone may have would be greatly appreciated! Thank you!