When you want to use a MH bulb in an HPS ballast you must use a conversion bulb. It must be the same wattage as the HPS bulb
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it would be ok in a circuit which didn't need a light bulb.
It can, but if you're wanting to run a 120v light bulb on DC, you'll need 120v DC to get the rated output. That's a lot of batteries. It's easier, and more sensible, to find a DC rated light bulb, such as an RV bulb.
To make a simple series circuit to light a bulb, the simplest components are a power source (such as a battery); a switch (to turn the power on or off); the bulb (obviously !); and some wires to connect everything together.
The bulb may be faulty or loose, there may be a loose wire or the battery may be in need of replacement.
That will vary on the manufacture and quality of the bulb. There is no set amount of lumens for a 40 watt halogen or any kind of bulb for that matter. To find out the lumen output you need to look at the box it came in as it should be stated. If you don't have the box it may be tricky to find out. You will have to see if you can find the manufacture somewhere on the bulb then google it and check the specs. It's probably safe to say its going to be around 500 lumens on average.