Your diagram shows a 2x4 as your nailer (ledger board). Do not use a 2x4 go larger 2x8.
Now the diagram is not saying this but just by eye it looks to be depicting 2x4's as the rafters. Again,
do not use 2x4's go larger. You might get away with 2x6's but given the snow load and load calculations for dimensional lumber based on 16OC it calls for 2x8's.
Are you going with pressure treated posts? I would not worry about warping, using a 6x6's is fine but you can get away with 4x4's.
As far as the roof pitch goes you generally want at least a 4/12 pitch to shed water properly but you are not going to get it with these dimensions.
Now I am not an expert but I think your pitch is going to be something like 2/12. As I understand it you would be better served with a metal roof for this pitch.
The real question is do you need a permit and will there be a building inspector involved. If so, you need to stick to code to pass.
PS to add. Have you given any thought about how you want it to look. Is it going to be just a functional screen porch or do you want it to be aesthetically pleasing. Are you going to have exposed rafters. If so, you might consider some nicer looking wood,a 2x8 looks like crap unless you paint it.