1. You can get 1:1 for limited like 120-160 degree. I use Opentrack so you have to increase the scale of axis, but you won't be able to look back, as you will lose tracking I would suggest keep it just like now.
1.2 Tighten your clip to your VR Headset with elastic or some thing else for comfort and better tracking
1.3 Yes you can use gyroscope of the phone it will give you 3DOF. Use Opentrack and select PI tracker(It will install PI tracker app on your mobile as well). Also if you do a little bit R&D on Trinus. There are plenty of headtracking options. One of them is using mobile gyroscope. I haven't tested it.
2. You need to use Riftcat. It will enable you to use Steam VR API
3. Yes people are using moonlight. I haven't tried it myself but this is already discussed in this thread or in the forum