4'x8' coop advice

[Urbanforager]

Structurally it would be as strong. However by screwing them together you run the risk of water seeping between them and rotting out the wood.
I think those 2x6s are pressure treated but the picture is a bit blurry so it's difficult to see the wood color. Even so, you want to protect the wood from contact with water. Especially end cuts.

There all treated 2x4 was planning on staining/painting everything cuz the lady wants a "pretty coop" if ya can't tell I don't have much experience with woodworking besides building ramps when I was young

 

[mowin]

If it all PT, I'd just make my own 4x4's.
Your going to need a bunch of 2x4 to frame out the walls, windows, doors ECT. Not sure where you live, but if snow load is a issue, I'd use 2x6 for the roof.