effiency of generator and house A/C vs dash A/C

If you run the generator while driving what damage if any, occurs when the house A/C cycles off resulting in no-load on the generator??

I don't agree, the engine A/C is fairly efficient.

As posted above, the box will be cooler if you run the house A/C. I will sometimes start the generator a while before stopping to get the house cooled down and if it is really hot will run the generator all day when needed.

Are you talking about overheating the engine while using he engine/cab air conditioner? That shouldn't be an issue.
But, it's a good reason to run the generator and house A/C if it is a concern.

I don't think it's any more or less efficient running the generator/house air. The difference would be minimal either way.
Don't worry about the fuel use. Fire up the generator and be comfortable.

You will not hurt the generator by running it when you are driving.

I don't know about more efficient but it certainly keeps the box cooler with the genny and roof top ac running. I will use both when it is real hot out. It's nice pulling into a park for the evening and the coach is already cool.