I don't have a run-in shed, but a pole barn.
My hubby designed and then we made the barn with stalls under the roof ourselves.
That said, we did a lot of online looking at ideas others had and then tweaked a few to make what we now have.
One place I found valuable for ideas different and the same, but made to withstand some serious horsing around was Horizon Structures. https://www.horizonstructures.com/horse-barns
Looking at their workmanship, construction and how this worked for that was amazing and idea-inspirational.
The types of materials they used, and combined together to make function also eye-appealing and long-lasting gave us things to think about, some we incorporated and some I've added to my future wish list.
I don't think this company has any structures in your area but you are building yourself and just looking for ideas...ideas they share willingly on their website.
The few pieces of information I will say to consider for your shed is a overhang and either insulate or place plywood of some wood product under the metal roof for several reasons and gutters.
That overhang by itself will make a huge difference in year round accessibility for your horses, having it be friendly in rain, snow, heat and cold weather times for them and you.
Any barn I ever do again will have a huge overhang as I have found how much my horses enjoy coming "home" daily to rest under the quiet of the barn security yet not be inside...
They are protected from all the weather as they choose and often just hang-out during the day to escape sun, flies, heat...forget the rain.
With that overhang we also keep our stalls dry, the horses protected from driving rains..mine don't get wet!!
Best is to get up early AM and see my horses all lying down under the overhang resting...
My neighbors horses when I invite them here, first thing they all do is drop and lay down under the overhang and rest...there must be something about overhangs is all I can think...
So the roof material itself...
Plan on some kind of under-layer or insulating of the metal roof...
My pole barn is metal roofed...still looks nice years after building it even in the Florida weather which is punishing...
It is horribly noisy in a downpour, so loud you can not hear yourself think "LOUD"
My horses going under the barn at night collects condensation that once morning sun arrives now drops off onto anything below...tack, feed, horses backs, stalls and shavings...all can get pretty wet when dripping occurs for several hours till the sun dries up all that residual moisture...then we repeat it the next day.
We have a full-length cap to allow heat to rise and leave.
I wish we had known to insulate or place sheathing under the roof to insulate from heat/cold and to quiet the noise factor of rain with our weather daily having torrential rains in rainy season.
Otherwise, we love the roof, the brown color has withstood the onslaught of Florida weather extremes of punishing sun daily.
In fact, when our home needs a new roof done, hopefully many years from now
, we will be having applied a metal roof as it can also insulate further our attic space reducing the overall heat built-up we then need the A/C tubing to cool down to pass cooling air throughout our home.
Nearly zero maintenance and the gutters/leaders already on our home can still be utilized so the deluge of water from storms is dissipated not puddling at the foundation.
Yes, regardless of where you live, any barn to me should have gutters installed to channel water away
from the barn to reduce mud, standing water you need to traverse and flooding that often happens when horses pack the dirt making trails the water then follows in making the barn area wet.
A run-in shed if mine would be sporting gutters too for the same reason.
Enjoy your project and some pictures shared please.