From the looks of it, you are correct with the descriptions. I can't see the colors of the valves (your wheels), but it looks like it's the correct descriptions based on position.
Has the system worked at all this year?
Do you have a backflow prevention assembly that has been turned off?
Does the Hose come on when you open the hose bibb?
Are there any other valves in the basement, or is this on a slab?
The main valve should be on all the way, then back it off 1/4 turn.
You said that you can hear water running when you manually turn the irrigation valves on. Where is the water going? Leave the valve on for several minutes to find out where the leak or problem is. If you can hear water running, you have to find out where it's going.
You don't have to be there first thing in the morning to see if the system comes on. You should be able to temporarly program a start time for when you are there to see what the problem is, or have a manual zone start from the controller.