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Differing opinions here, and thats a healthy thing...
By putting the run below grade, you are encouraging problems. If the concrete is above grade, it will form a dam, holding upstream surface flows out of the run. IMHO, the ring of concrete would need to be higher than the surrounding soils, and sloped at approximately 2-4% toward the outsideof the run. This is the normal slope for drainage. By sloping the concrete to the outside, you end up with the gravel run raised above the surrounding soils. The chickens will want to dig in the gravel/sand so you will end up with some pitting. This means you need to carry the 2% slope up into the gravel, but know that whenever you rake it out, you will need to re-establish the grading to prevent ponding in the run. Basically, the run and apron would endup looking like a small hill if this is done right. Btw 2% is the slope across roads and sidewalks so that they drain properly. Since you are using the same materials, you should expect the same drainage rates and patterns the roads see.
As for the concrete, as a general rule, the dryer a concrete mix is, the stronger the concrete will be. Whether the concrete is mixed in a mixer or a hole makes no difference as long as the right amount of water is used in the mix, and the dirt from the hole doesn't get mixed into the concrete. Since th econcrete will absorb some moisture from the soil, I would suggest reducing the water used in the mix by enough to offset the moisture in the soil. This is where things get tricky. If you can do the math to figure out how much water the concrete will absorb, I want to hire you. Reality says though, that even without any concrete, the fenceposts should hold OK as long as they are set deep enough. Many fences are built without concrete to set the posts. Setting the posts with a bag of sand can actually be a better system than concrete since the sand will allow for movement without being thrust up by winter frosts the way concrete is.
I would use the screenings for post setting, however I would not use them for a run in a rainy area. Screenings have a higher content of "fines" or dust, and they will pack down to form a barrier that the water has a hard time getting through. A coarser gravel should allow for better drainage, and still allow the chicks to dust bathe/scratch. If Screenings are to be used, I would suggest a thin layer over a bed of coarser gravel, graded to around 2%.
Just a couple suggestions from a guy who has never built a chicken run.