Wood – How to correct a twist in a 2×4

2 2x4s are not square, if that matters (3 x 3.5). They will be plenty strong as assembled posts.

Try a traditional lumber yard, they will have untreated pine. Cedar may also be an option.

Your structure would be better supported if you shortened 1 of the 2 paired 2x4s at each corner and made a jack-king stud arrangement like a door frame and header. The main tenant of construction is to bear the weight with structural components and only use fasteners to keep the structure in place, not to bear the load (in shear).

You are, in effect, making a small deck. You should attach the rear 2x8 2x6 to each wall stud. A 4" #10 decking screw would not be too small .

The front and rear members are, in effect, beams and should be 2x8s. The joists running perpendicular to the beam should be 2x6 2x4s hung with joist brackets.

Plywood will not sag over time if you intend to use it for the door faces. I recommend birch plywood for not only its hardness but looks. However your hinges and what they are attached to are the big concern in this project. It looks like you've used a low grade pine to build the unit. This is a soft wood and a couple of cabinet hinges screwed into this are going to loosen quickly. Your best chance for longevity is to use a piano hinge so that you have a continuous hinge from top to bottom with 15 screws or more on each hinge. I highly recommend ordering a stainless steel or flat nickel piano hinge for this. What they have in stock at the box stores is of the "bright" variety and looks cheap to me.

I'd also recommend that you not attempt to build the door to be perfectly flush with the "lip" at the bottom. Leave yourself 1/4 of an inch gap. Units like these always sag at least to some degree and you wind up with the door scratching the "lip" and its going to look really bad with time.

Also take a look at ball catches to use either at the top or bottom of the doors to hold them in place once closed.