Wow, you have quite a few perspectives in responses so far. (That's good.)
My 11-yr-old has had this kind of reaction to various items (and situations). It almost doesn't matter how "attached" he is. It just seems that, in the moment, it appears to have an "end-of-the-world" grip on him. He can't let go for quite a while.
He is diagnosed with Asperger's, OCD and anxiety. His medication helps a great deal so these meltdowns happen a lot less now (maybe 4 or 5 times a year instead of several times a month).
So I know this can happen to some kids. My son's pediatrician wanted to put off sending him to a psychiatrist/psychologist but, with my gut/Mom instinct, I spoke up that I my concerns were big enough to ask for more help. His reaction was, like, no problem; I'll get the referral set up. Remember that you have a lot closer relationship with your son than the doctor does and what you see in your son is hard to describe well enough to share. So don't hesitate to speak up.
I would recommend that you don't demand extra chores or other ways to help pay for a new coat or jacket. Take him to the department store when you can and show him all of the possibilities out there. He will probably (I'm guessing, with what I have seen with my son) find that he would really like a jacket that fits him and is something you can afford.
Also, I would recommend contacting his aunt and uncle and find out 1) if the jacket is in their vehicle; 2) if they have a way to call about lost and found items at the football game location or can go take a look when they have time. If found, either stuff it in a flat-rate box and pay $4.90 or so to mail it or take a weekend drive for half-an-hour and meet them halfway. I have done this with a DS game which my son really loved to play but accidentally left behind at grandparents' house which is two hours drive away.
Something I would consider a situation where I ask my child to help pay back for it would be like my 16-yr-old and his headphones. He uses the more expensive game headphones because of the built in microphone and he is tough on them. I can tell you that my husband and I make him save up his allowance to buy replacements. After two new ones were ruined or broken within a couple of months of use, my husband and I just plain got tired of such lack of respect.
I, too, think there is transference of his loss onto these objects. Continue with the self-esteem building. He needs, it seems to me, extra reminders that he is not alone and is loved very much.
I think the idea of some breathing exercises was brilliant and be ready to spend more time with that. Lori Lite has some way awesome music, with meditation-like exercises and affirmation exercises for kids. I picked up some from Amazon and really like them.
When (if) your son won't let you help -- just doesn't want to listen, encourage him to draw or write. Don't direct the drawing. Just make it something to do. He will express himself without really realizing it and maybe that would be a help with calming him.
Let us know what helped/helps! Good luck,
D.