i DO NOT MEAN THIS TO BE SNOTTY AT ALL, SO DON'T TAKE IT THAT WAY, BUT IF THIS WERE YOUR FAMILY WOULD AMOUNT OF TIME FOR AN EXHAUST THAT CAUSES CANCER COMING RIGHT INTO YOUR YARD, DOORS, WINDOWS, AREA OF MOVING ABOUT ON THE PROPERTY BE OK ?
As a matter of fact, my own neighbor has their driveway 5' away from my bedroom window. They occasionally idle one of their cars, and yes the exhaust is not healthy. But it is only occasional, so it's no big deal. Likewise, our own gardeners fire up their gas-powered leaf blowers every Monday morning. Because it only occurs once a week, we make sure all the windows are shut at that time.
When emergency generators and the emergency systems they serve are being installed, they are often tested for extended periods. After that, the use usually drops way down, to save on fuel costs. Have you heard whether this neighbor intends to run the generator constantly once the building is occupied? Of will they only run it in an emergency or as part of a monthly test, in which case that's not much different than your neighbor occasionally idling a vehicle near their property line.
Aside from that, we are telling you to take the building permit out of the equation, as once the generator location is determined to be code compliant, any discussion about use is beyond the scope of the building construction inspector.
Think of it this way: If one day your neighbor was burning tires next to your property line, would you call the building inspector? I suspect you would instead call municipal code enforcement, or perhaps call your councilperson's office; maybe they would check with the planning (not building) department to see if that was a permissible use.
If your neighbor was playing paintball and they shot paint all over your roof, would you call the building inspector? Or would you call your insurance company (and they might make a claim against your neighbor's insurance company), or maybe call the police if your think it is a type of vandalism? (Making an analogy to the concrete you said the got on your roof.)
Here in California, we have additional environmental regulations that (no surprise here) you may not have in Texas. Every new development is subject to the California Environmental Quality Act (CEQA). If the environmental impact study did not take generator use into account, you could potentially sue under CEQA. If the generator was not approved by the Air Quality Management District, they could also provide enforcement.
My point is, you may have an environmental issue, but not a building code inspection issue.