D.B.
I think your mom and aunt must have watched a TV show or read an article, and now they are trying to diagnose a toddler. That panicked you, and you headed to the internet, and that panicked you more. There are a zillion reasons for circles under the eyes (allergies, lack of sleep, an oncoming cold, etc.). Since he gave up naps, it's very likely that there will be a period of at least 6 months when he's abnormally tired - kids usually give up the naps many months before they are truly ready to!
The bruises your child displays are called "having fun" marks by many pediatricians - very common in active children. By themselves, they indicate nothing.
It does sound like your child has a limited diet so you need to either sneak nutrients into his food or you need a comprehensive supplement that not only provides iron but a lot more. I work with a lot of kids with sensory issues so we see this a lot. Fortunately there are some delicious ways to get a children's formula into him to give him protein as well as many other nutrients (not just iron or what's contained in typical vitamin products sold in supermarkets). You can't just give iron - the body won't absorb it in the absence of other nutrients. Same goes for each individual vitamin, calcium, other minerals - you can't give them individually just because a person tests "low" in that one element.
So yes, get him checked out, but don't panic. You can look at medical info or food science/nutrition or both, but don't assume that your mother and aunt are authorities, and don't assume that a deficiency at birth is still present now.