Honestly, there is no best way. Rows are good, square foot is good. Raised beds are usually best of you have drainage or root issues. Don't look like that is the case in your veggie area. And remember, you can start small. Plant you veggies now and expand your veggie garden later.
Why are you using roundup? If roundup drift gets on any of your "good" plants they will be gonners. Seems to me you are doing fine just digging up your grass for the veggie bed.
You are very hung up on your chart of sun hours. Honestly 4 hours of sun not much for any type of veggies. Probably the bare minimum for leafy greens. I don't know why you are trying to reduce the sun unless your climate is too brutally hot in the summertime. People plant a garden in a sunny spot, they usually don't try to reduce the sunlight much.
And some things you can not change. Bok Choi, Tatsoi, lettuce and most greens are what are called cool season crops. They will not grow in the heat of summer in sunlight or shade.
