Once the immune system is compromised, you're susceptible to everything else that comes along. Once the body's balance is upset by an infection (viral, bacteria, fungal...), it's open season for other things. The antibiotics can cure one thing and then throw everything else off (which could be the cause of the diarrhea, with "help" from increased mucus from postnasal drip from the allergies). I'm not sure it's a question of not trusting the doctors - it's more likely that you aren't used to the allergens in the air, and the kids just don't have a strong enough immune system to beat these things. They're getting something new before they are recovered from the prior problem, plus they pass things back and forth. Isn't it nice how kids who can't share toys can manage to share diseases???
I work in food science and immune system support, and our family members are all illness-free and allergy-free now, and have been for years. So you can go that route, or you can just increase the rest and fluids. Everyone will tell you to "eat right" but the AMA wrote many years ago that it's virtually impossible in this country to do that, with depleted soils and reduced nutrients in foods (even our fresh foods are down as much as 40% in nutrient content). So you have to supplement intelligently and comprehensively (not just a little multi-vitamin, which doesn't begin to replace what's missing from our foods), or you have to deal with the hassles of recurrent illnesses. But this is not a medical issue, it's a food issue. Some doctors are educated in this area, but most are not. Our medical system is highly geared toward acute illness, not chronic illness.
But I do think you can shift away from losing your mind with worry, and just go directly to "impatient and frustrated." This is very typical no matter where you live, and I see it in my work all the time. I don't think there's much the doctors can do for you. I would urge you NOT to use any antibacterial soaps and cleaners whatsoever (just regular soap and more natural cleaners like baking soda or vinegar), and let your kids play in the dirt.