Rather than having 2 4 purlins every 12 it would make far more sense economically to use 2 6 purlins every 24.
Ok to use 2 x 4 for for roof rafters.
As mentioned above a 2 4 spf shed roof rafter can span 11.
At the pitch range you are looking at the 2x4 is just fine up to 8 6 depending on the lumber grade.
The back of the roof will attach to house with 16 2x4 and lag screws.
Front and sides will sit on deck on a 2x4 wall with studs 24 oc and screened in.
If your project had a rafter length of 8 or less a safe and strong method is to use 2x8 framed 16 on center.
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.
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.
It is always better to have a large rafter rather than be concerned with a sagging roof.
Makes construction easier too.
The secret word is truss.
Also if you want to make things cheaper you can do a single slope of the roof and basically cut out the eve cost.
You can use them for roof floor and even wall framing but few sheds will need trusses for walls.
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.
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.
Rafters to 2x4 with joist hangers.