You can frame a whole house with nothing but 2x4 but it s easier to do a shed as you won t need engineering probably.
Ok to use 2 x 4 for for roof rafters.
The back of the roof will attach to house with 16 2x4 and lag screws.
Somehow i am in a traditionalist camp 16 oc for a 12 foot wall will add only 6 extra 2x4 s or 3 extra if you buy 10 foot lengths for the rafters and 6 extra 2x4 s for studs if you use only a single layer top plate on the bearing walls.
Also if you want to make things cheaper you can do a single slope of the roof and basically cut out the eve cost.
Structures such as sheds or barns can be frames with rafters every 24 again make sure to consult local codes.
Makes construction easier too.
If you live in an area that requires a shed roof rafter to support a heavy load then an spf rafter can only span 7.
It is always better to have a large rafter rather than be concerned with a sagging roof.
At the pitch range you are looking at the 2x4 is just fine up to 8 6 depending on the lumber grade.
The secret word is truss.
As mentioned above a 2 4 spf shed roof rafter can span 11.
Rafters to 2x4 with joist hangers.
An syp rafter can span 11.
Less expensive 1 3rd less board footage of lumber fewer pieces to handle fewer joist hangers to have to attach and only one half as many screws to attach the roof steel.
If your project had a rafter length of 8 or less a safe and strong method is to use 2x8 framed 16 on center.
Rather than having 2 4 purlins every 12 it would make far more sense economically to use 2 6 purlins every 24.
Both of these are for light roof loads.