After switching on the same system from Unity, compiz and lightdm to gnome3 and
gdm3, updating drivers (oibaf ppa) the remaining issues are almost gone.

1. I do not notice not drawing content in the windows issue anymore.
2. Tearing is still visible in Steam app, but it is right now just a blink from
time to time, hard to notice.
3. Haven't noticed any other heavy tearing and flickering.
...
Rest points were solved previously. I remember I had a lot of tearing while
playing SeriousSam3 and STALKER (through wine) on DRI_PRIME=1, will have to
check it out again with gnome3 desktop.

If everything will be fine with above, might compiz cause the issues? Or is it
because I've up-to-date LLVM and Mesa drivers?

glxinfo:
Gallium 0.4 on AMD OLAND (DRM 2.43.0 / 4.4.0-81-generic, LLVM 4.0.1)
3.0 Mesa 17.2.0-devel