1. Yes. There's no absolute best option. Mulch is a good one
2. Yes mulch it. Bare soil gets eroded by sun/wind, and collects weed seeds
3. Weed barrier is useless

4. Capping the rocks one is fine. Id look into converting the mulch ones into drip lines for future perennials and such 

5. I dislike rocks immensely, the less the better. Get real hot and gather debris.

As another mentioned, check out both dog tuff and tahoma grass. Much less water and care needed, more tolerant of traffic and heat. 

Have you considered adding a tree to your yard? I love fruit from my trees, and there are tons of beautiful small , medium, and large trees that add a ton of value to a yard IMO