Yes, everything that Bokonon said. That's the key - formula feeding entails certain risks. It's not that the breastfed baby is healthier, it's that the formula fed baby has a higher risk of. . . respiratory and GI illness, allergies, asthma, diabetes, heart disease, cancer. . .it's a long list, and it's lifelong.
When we talk about risks and benefits, we're talking about populations. There's no way to know for a particular individual what the effects of not breastfeeding will be. This is also why some folks claim that they or their children were formula fed and are "fine" - that person STILL has an increased risk of all of the above, but what is their baseline, genetic and environmental risk? If it's low, then they might be okay. If they have particular risk factors, then they might be really in trouble. And so many of these illnesses have many factors that affect them, in an individual it's impossible to parse out a "cause", so if an adult develops heart disease you can't say it was or wasn't due to the way they were fed as an infant instead of the fact that they ate McDonalds hamburgers 5x a week for 20 years. (You hear the same misunderstanding in statements like, "My grandmother smoked 2 packs a day and she lived to be 90, so cigarettes can't be that bad.")
More than likely, your DS was sick less often and less severely as an infant/toddler then he would have been if you'd fed formula. If he has allergies and was breastfed, likely you wouldn't be mentioning "probable" allergies, they'd be definite. My DD1 has eczema (severe while she was still breastfeeding), seasonal allergies, and now it appears asthma as well. I don't think that's a failure of breastfeeding, it's just her genes, which unfortunately in this case are my fault, LOL!
I was so smugly satisfied about breastfeeding with my twins who were never sick as infants and toddlers. Well, my 3rd child is sick all the time. She had a cold a month for her first 6 months. Exclusively breastfed, never a drop of formula (which the twins had), but the same circumstances as your DS now - in a school-aged community, exposed to other children on a daily basis, kids are going to get sick. That's just the name of the game. You can reduce the frequency and severity to some extent with frequent hand washing, teaching kids not to touch their eyes/noses, eating a healthy diet, getting plenty of rest, but even w/ all of that they will get sick. And darn it, I was breastfed but I always get whatever any of my kids bring home and I'm always the sickest - how's that fair?!? 
As far as behavior, sounds like a typical preschool or early gradeschool boy to me. So much of that is innate personality and/or age-related. Breastfeeding a toddler or preschooler can help with connection and awareness of your child's needs, so it can help you manage those types of issues, but it isn't going to make an extrovert out of an introvert, or prevent a tired, upset child from tantruming. There are some great resources for positive discipline and working with your child's personality that might make it easier.