Software that tells for how long the computer has been running on battery

Actually, an Apple battery gives you the percentage of its original capacity that it currently has. You can access that number through the free CoconutBattery app.

If you want to do your own benchmark, I recommend using BOINC and supporting a grid computing project, like SETI@home. These basically use 100% of your CPU from the moment you turn the program on until the moment you turn it off. This drains the battery pretty quick. Just be careful to check temperatures to make sure the computer doesn't overheat, especially with a MacBook Air. You probably should quit extra apps, but it shouldn't matter too much.

Yes, you can achieve that using certain EFI variable. I answered this other question with the full procedure to permanently disable discrete graphics card on startup.

Preventing your computer from switching to discrete graphics card will allow you to asses if that is the reason for your poor battery performance.