I’m not even going to try to come up with a maximum number of chickens you could squeeze in there. My main coop is half the size of yours but my main run is 12 x 32. That’s still smaller than yours but not tremendously so. 15 x 38 sounds big on paper, and it is compared to what a lot of people have, but on the ground it’s not huge. I also have an area about 45’ x 90’ inside electric netting. I have large hawks all over the place but have never had a problem with them, even with young chicks in that netting area. Predators can be funny that way, certain ones a problem for some but not for others. I sometimes have over 40 chickens in there but many are young, just growing to butcher size. And I have a grow-out coop and another smaller coop I sometimes use to relieve the pressure on my main coop. My set-up is different to yours but I think I can sort of relate. I’m sure we manage them differently. We are all unique.
You can follow the link in my signature to get some things I feel are important about chickens and space, but a few things I consider pertinent to your situation. What is your flock make-up? How many roosters versus how many hens? Only one rooster doesn’t do much to space requirements but each additional one you add can make the space available look really small pretty quickly. Two or more roosters can get along but often they split the flock and each has its own territory so they avoid a lot of contact. If their territories are within line of sight of each other, well sometimes it can cause problems. Sometimes. With all this stuff, since you are dealing with living animals no one can give you guarantees. About anything can happen.
How often are you going to be integrating new chickens or have broody hens raise chicks with the flock? A broody hen needs a certain amount of room to work, but the real need comes after she weans them or when you integrate brooder-raised chicks. They rank at the bottom of the pecking order and will form a sub-flock, avoiding the adults as much as they can. Having sufficient room to avoid the adults is pretty important in this phase either being pretty peaceful or it getting messy.
Poop management can become an issue. Some of that depends on your climate (mainly how wet it gets at times), your soils type (how well it drains), and the slope of the area, including the area around the run. Water is your biggest enemy in this. The more chickens you have pooping in a certain area the more it builds up. When poop gets wet and stays wet just a few days it can stink, plus a wet run is an unhealthy run. If your run is on a slope or, even better, on a hill top so water can run off instead of drain to or through the run, you are much better off than if it is where water drains to and sits. Sand drains much better than clay but the water needs someplace lower to drain to. Even with just a few chickens a wet run can become stinky but if you have a lot of chickens pooping in there the effect is magnified.
To me the best run is one where it is built up so any water that gets in drains out. Poop is pretty much water soluble. The gravel idea could work provided the poopy water gets out of the run and doesn’t form a wet lagoon nearby that is going to stink itself. Feedlots often stink and often have special water treatment lagoons set up to help treat the water. Sometimes flies are an issue. That’s because of poop concentration. Packing chickens in very tightly makes me concerned that poop management may become a big issue for you.
There are all kinds of ways to build a run. For one that size and especially uneven ground I’d suggest fence posts and fencing. Your corner posts probably need to be pretty substantial, maybe even braced, but the intermediates could be T-posts or something like that. Remember to make the post you hang your gate off of pretty substantial and brace it if you can at all. You don’t want a sagging gate.
With your easy to dig in ground, you could dig a trench and bury the wire, but I’d suggest an apron is equally effective and a lot easier to install. Either bend the bottom say 18” of your fence at a 90 degree angle out or attach a piece of wire 18” to 24” wide at the bottom. A digging predator goes up to the fence, digs, hits the fence, and doesn’t know to back up. You don’t have to bury it, grass will eventually grow up through it, but if you remove the top 2” of turf and put that back on top you don’t have issues with weed eaters or lawn mowers. That should take care of any uneven ground issues.
What wire to use? That depends on your risk tolerance and your pocketbook. A decent sized dog, a big raccoon, or a coyote can rip most chicken wire. Different chicken wire comes in different gauges, the smaller (heavier) the gauge the better but more expensive if you can even find it. I used 2” x 4” welded wire and covered the bottom 18” with chicken wire. That will stop most larger predators (not bears) but different chicken wire comes with different sized holes. Snakes, many weasels, and rats can still get through most chicken wire.
One problem you may have with your run as you described it is that the top might sag too much. You said nowhere would it be less than 6’ but are you taking sag into account? Can you walk under it without stooping? You may need to brace it in the middle. If your only concern is hawks and keeping the chickens in the wire you use for the top doesn’t have to be that substantial, but a wet snow or an ice storm can cause it to come crashing down. I don’t know the best way for you to brace it but I kind of envision a couple of posts in the middle to help stop the sag. Maye your sides are high enough that this isn’t an issue, but if it is, a brace later may be an easy fix.
I like Lazy Gardener’s idea of sectioning off parts for separation. I’ll even suggest you consider making that small outbuilding a “grow-out” coop with a section of the run fenced with chicken wire that can be isolated from the main run and put your storage in the main coop. Mine is set up that way and I find it really handy during integration or when I isolate a bunch of cockerels while growing them to butcher size. That flexibility comes in really handy. I normally butcher my cockerels after five months but wanted hatching eggs from my main rooster so I isolated the cockerels in there to be sure the right rooster was fertilizing the eggs. You never know when a little extra flexibility will come in handy.
Hopefully you can get something useful out of all this typing. Good luck!