Baby with eczema

Hi there, my 4.5 month old baby has had eczema (at least, that's what the doc says it is) for about 2 months now. It began looking like an oval, red rash that was just a few spots on the underside of his thighs and then progressed to encompass his legs and arms. I keep a homemade herbal oil on it, and the red has died down, but the skin is a bit scaly and very dry. He gets a "drool rash" on his hands from them being constantly in his mouth. Red, chapped skin there. His cheeks tend to be very dry and from time to time are reddened and warm. He is a big baby, in the 98th percentile in height and weight. Eats very often, is breastfed. I feel that his underlying personality is placid and cheerful, laid back and cool as a cucumber. And so he was at birth. Calm, hardly cried. We thought something was wrong with him, he was so quiet! You'd hardly know there was a baby in the room. But gradually since then he has gotten more demanding, and less content to sit and plat happily. He gets grumpy and will fuss after about 10 minutes or so after setting him down. He's bit terrible, but I feel he could be better, that he's a bit too easily irritated and easily bored (wants distracted a lot) compared to how I know his inborn personality truly is.
When I was pregnant with him, I had a dream that he was just a calm and happy newborn, and he turned out to be just that. I just worry that he's getting imbalanced. He is still much happier, cheerful and calm than my older two are by comparison. He laughs and giggles and smiles a lot.

His tongue has had a thick, white coat since my milk came in around day two of his life. I kept asking the docs if he had thrush, but they don't think it's that.

His feet and armpits tend to smell sour when it's been a few days since he's had a bath. Seems like he sweats on hands and feet most. The skin on his legs, arms, and face is dry and chapped, despite frequent application of herbal oils. (Chamomile, oats, and yarrow infused in coconut and sunflower oil mixed with cocoa butter).
He has a slight bit of scabby crusts behind his ears, in the creases there. He has tons of earwax. I feel his ears need cleaning on almost a daily basis. He has some cradle cap, but not very noticeable unless you look closely.
His skin seems just a bit pale. I would peg him for the "phlegmatic" type, or the Kapha, depending on the model you use.
He drools copiously, soaking his shirts. No teeth seem to be erupting.
He almost always has his tongue out and rolling around, like he's playing with it inside his mouth or chewing on it.
He has lately been rubbing his eyes a lot, especially when tired, but excessively so, as if it itches or bothers him. Seems he rubs his right eye more than the left, and his right cheek seems redder than the right when his face is flushed. Also since birth his right eye looks just slightly wider open than the left.
Sometimes at night he will wake up screaming, usually because he has gotten too hot under the covers. Yet, at birth, he had a hard time maintaining his body heat, and had to be in the warmer for an hour or two. I try to strike a good balance between warm enough but not too hot for him.
When he gets sick, he usually makes tons of nasal discharge, and in fact is still getting over the last cold (been a couple weeks now). He still has green discharge, thick and sticky. During illnesses he gets coughs which sound phlegmy, but that never go all the way down into his lungs, just rattles around in the upper airway and sounds bad without being too serious.
He did have a course of Prednisone around 1 month old to help him with a bad cold he caught at that time. It never got too serious then, but we were nervous and watching him very carefully with being so young.
He wakes up about 5 or 6 times a night. This is something I wish he did better on, so i could get more sleep. I have been sleep deprived by all my kids, the oldest didn't sleep through the night until about 4 years old, just before the middle one was born, who at almost 3 years old still tends to wake up and need me once or twice a night now and then. So I am really getting burnt out and could use a happy baby who sleeps through the night, or who is at least moving closer to that possibility.