Classic allergy symptoms ! Talk to your pediatrician about getting something for him for seasonal allergies. The dry spot is probably from rubbing,(since he is rubbing on the sheets). It could be eczema, but I doubt it because you would have described it more as a scaly, or dry red rash. Moisturize it.
The general rule of thumb for rashes is this;
if it is moist, dry it up (such as with cortizone), if it is dry, moisturize it. (Such as with baby lotion or baby cream)
He is probably teething too, since he is 5 months old, but excessive drooling is also a classic symptom of allergy issues. If it were my child, I would definitely be asking what I could give him to relieve allergies. They have a lot more products out there now than they had in my day, so I won't recommend anything. You'll have to ask your pediatrician to do that. One thing I WILL recommend though, (and it can be used even with meds) is "Little Noses". It is simply a saline nasal spray, to keep the nasal passages moist. That cuts down on some of the itching, and helps to prevent sinus infections. My daughter has been using it for my granddaughter since she was a tiny baby. It is so effective, that I carry a bottle in my purse for my own personal use. (I, too, have allergies, asthma, and my sinuses can get terribly dried out from medications) It's great stuff, it's over the counter and it's cheap.
I wish you well. :o)