Patio gate not at right angle to the house

Great question. Adding this addition requires a really good, deep, compacted base. I would isolate it from the building by 1/2 inch, but if the base is good, rebar to the existing pad will help stop any displacement. You have to expect some settling, but the better the base, the smaller the amount. Web rebar won't help much, but 3/8 or 1/2 inch bar will hold the joint. The trick will be compact, wet, compact, wet. Make it rock hard. Good luck.

I recently saw someone set a length of galvanized pipe in a concrete footer, and then bolt a pressure-treated fence post to the metal pipe. The galvanized pipe and footer should last essentially forever, and provide plenty of strength to support the fence/gate. The fence and gate can be replaced as needed. I thought it was a fairly clever approach. You might consider it for your application.

It's perfectly normal to have two gates without a center post, using a ground pin (often called a cane bolt) to hold the less-often used side of the gate stationary. The only downside here is that the gate will twist a bit more if side loads are applied near the top of the gate (e.g. wind, someone pushing on the gate), since the only fixed point is at the ground.

The longer the cane bolt, and the further up the gate it extends, the more support it can provide to the gate. Taking this to the extreme, if you were really concerned about the rigidity of the swinging end of the gate, you could make your own "cane bolt" out of a long section of pipe (perhaps 4 feet), and have it drop into a larger pipe set a foot or so into the concrete. That gate wouldn't go anywhere!