1) Run the 2x4's under the roof vertically, not flat like they appear to be. It will give the roof a lot more strength. (Any kids who willl think it is a fun thing to play on?) Otherwise it will begin to sag if you do not have a load of wood holding out the sides.
2) The 2x6's are not needed at the ends, you can use 2x4's. You would be better off running 2x6's under the roof from end to end if you are concerned about strength. (Kids?, sagging)
3) You can use 2x4's for the stringers. I'd make the center vertical a 4x4 (or pair of 2's nailed together if it is cheaper) Having the 2x stringers spaced in a little will be a lot more forgiving when stacking odd ball sizes of wood, instead of having to share the nominal 4" of a 4x between two stacks.
4) You can nail the green cross pieces as blocking in between the stringers. It does not need to go across the top.
5) X brace the ends also or turn two of the 4 angles 90 degrees, parallel with the ends, that will reduce front-back sway.