I would start to do a food diary and write down any reactions and how long they took to show and how long they lasted for.
I am a bit confused on this bit
"Same thing, hives around his mouth. As soon as I wiped his mouth off real well though, they'd go away. I haven't tried giving him eggs since. He seems ok when he has them in baked goods though "
If it was an allergic reaction wiping his face would not make them go away.
Similarly with a reaction to a food such as eggs, he would either show a reaction to them or not it would not really make any difference if they were baked with something else.
If you find that your child is allergic to a particular food, they are more likely to have allergies to foods in the same family. For example, if your child is allergic to a type of fruit for example a citrus fruit, he may be more likely to be allergic to other fruits in the same group (but not always!)
Children with excema can be prone to some food sensitivity but this is a small number and only makes the symptoms worse, it does not cause the eczema.
Eczema - Triggers and Irritants
This has a good section that explains how food can be a trigger factor.
There is a good section on AK site about allergies you might want to read, but my advise is a food diary until you see your gp.
Yes you can substitute soy milk/ almond milk in cooking.
Hope you get answers form your gp.