We have a free-standing pergola on our patio. Concrete footings with post stand-offs. Size on this one are 8' x 12' on post centers so larger than you are planning. Used 6 x 6 PT for posts, then skinned with cedar. Structure above is 2 x 12 however nested so the strength would be similar to 2 x 6.

Yes, 2 x 4 will be more likely to warp. Depending on your on-center, should adequately support lattice and shade cloth.

I'm linking ours in the assembly phase and plan schematics. Been standing 5 years now. Hope this helps.

Here is a link that might be useful: Pergola

Thanks for posting the pictures and plans. Very impressive pergola. I plan on doing something smaller just to shade a window and some plants that can't handle the Texas sun. I wish I could pour footings, but the structure has to be surface mounted to the existing patio. I think I'll use 2 by 6's and sandwich the beams, then make a sturdy lattice panel and put it between two posts on the end to provide shade and make everything more sturdy. Thanks again for your input!