As a city building inspector, I think I can shine some light on the inspector's responsibilities, as well as those of the plumber.
The fact that your water heater doesn't function has to be attributed to the installer or the manufacturer. If it is a manufacturer issue, your plumber should have caught it. If he didn't purge the air from the burner unit or light it correctly the first time it probably just didn't stay lit. If the problem persists, there may be an issue with the unit itself of the gas delivery to the unit.
Now, the inspector...
He's responsible for safety, not necessarily function. I'll cover function first. Because of liability, most inspectors will not adjust the dial on a gas water heater to determine that the unit lights. It would be an easy lawsuit for someone if they were scalded. Most cities' attorneys make this very clear to their inspectors. The residential plumbing code does not require that the water reach a certain temperature, or that it be limited to a certain temperature to prevent scalding. Tempered water (hot, but not hot enough to burn you) is in fact a requirement in most commercial construction based on accessibility laws and codes.
When I do a water heater inspection, I'm looking for the following:
- Is the unit stable? Will it rock back and forth?
- Is it vented correctly? Is the vent installed in a workmanlike manner? Is the vent too close to combustable products? Is the vent sloping uphill to facilitate good flow? Is the vent running too far horizontally?
- Is there adequate combustion air for the unit to use, given the space the unit is located in? If not, it won't burn as efficiently and will produce more carbon monoxide than usual.
- Is the water plumbing installed per code and in a workmanlike manner?
- Is there a gas shutoff valve and a union? If a flexible gas whip is used, is there a shutoff valve? Are all the gas fittings legal for gas use? If there are any new gaslines, are they sized and installed per code, and have they been pressure tested and approved?
- If there are copper or galvanized water lines, did they use dielectric unions or dielectric nipples? An 8 gauge copper bonding jumper must be installed to restore electric continuity to the homes grounding/earthing system.
- Did they install a water heater that is in good saleable condition? A while back a jurisdiction I worked for had a heck of a time with an unnamed big box (rhymes with crows) store's installation subcontractor. They'd install dented water heaters. I got about a dozen homeowners new undamaged water heaters, even after they'd been lead to believe that the units were just fine and would last as long as an undamaged one.
- Is the T&P blowoff properly piped down to the floor?
- I always ask the homeowner if the unit is functioning satisfactorily for them. If not, we can try to determine why.
There's a lot to look at, and I can do all of it in about 40 seconds without crawling around or making a big deal of it. When a homeowner is with me, I normally walk them through the inspection and show them exactly what I am looking for, just so they know the permit was worth obtaining. I've seen a lot of scary installations of water heaters that would have been guaranteed to get people hurt or killed.
I'm glad there's a good fire in your furnace! You're advocating for yourself! You paid good money for something that should work perfectly. Remember that the inspector is also advocating for you by requiring a permit. He's not making any money off of it, and could just as easily ignore the issue. Hopefully he's looking for all the things I listed (and anything I forgot to list but should have).
You've got the right attitude in dealing with the inspector, and probably have the right idea in asking that the plumbing company send you someone else to get it running. Perhaps someone that won't make Blue Angels jokes after you've spent a weekend taking cold showers. 
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