Those pre fab coops are darling, but really are limited to a few chickens. Your chickens are not full grown yet, so will actually be taking up more space. You just might be a bit too tight of space. You are going into spring, so the days are getting longer, so your birds will spend most of the time outside. The bigger run was a great idea. A home made larger coop would be a great idea too.
In the run, it is very helpful to birds that are penned in all day, to create multilevels. A pallet up on blocks, a partial wall in the middle, a roost in a corner, while making your run look a bit cluttered, actually increases the space in the run. An empty run, is basically a two dimensional space, and there is no where to get away from another bird. Where as adding a pallet, where birds can get under it for a bit of shade, and on top of it for a bit of sunning, actually doubles that space. I stick up a roost kitty corner and usually all summer in the late afternoon, a couple of my birds sit on that for the last bit of sunshine. Just make sure there are two exits, so you do not create a trap.
Pecking order is determined, by two birds meeting each other, and one bird leaving the area, leaving the dominant bird in possession of the space. In a run, there is no place to leave to, and this can lead to an aggressive bird, chasing and chasing a passive bird, because the bird won't leave. If a bird can go behind a wall, or under a roof, they can get out of sight, and all is well. You can actually keep more birds in a smaller area by using all of the space.
Personally, I would cover the run, I have lost lots of birds to predators.
MK