Head not Working: Obviously a problem. Given how simple that should be to fix (and how obvious the problem is) it would not be unreasonable to withhold final payment until fixed.
Leaking Value: You might be turning the top too much. While I don't know if all valves are this way, the few I've seen only require a quarter turn. But at least there isn't a leak when turned on by the controller, I guess that's what's really important.
Gravel under Valve boxes: It's a good practice to do. But obviously you don't have to to make a system work. I think that one is going to come down to what was in the contract. It nothing was specified, you're likely stuck as-is. Just try avoid walking on those boxes to avoid damage.
Head not level with the grade: According to
http://www.irrigationtutorials.com/instal16.htm 1/4" to 1/2" above the soil line is what you should do.