5 gallons of paint for 9 rooms isn't an unreasonable amount as long as the rooms are small too medium is size, have no texture (smooth) or a very light texture and the new paint is the same color. It will be cutting it close. So, I assume if any more paint is needed the painter will supply it as part of the job.
As for 1 gallon of water added to the paint, this is a little much for most paints. But, I have used some interior flats that needed some thinning just to make it work. 1 gallon, 20% thinning, is typical for painting acoustic (popcorn) ceilings where a thin paint is needed. I could see using 1/2 gallon of water for a thick paint going on a smooth non-textured ceiling.
You didn't mention how he will apply the paint. Excessive thinning can be "hidden" when spraying. For rolling, the thin paint will spatter more and be difficult to use.
At least he showed up to your house. I would never give a bid over the phone, to many variables. Some that come to mind are; condition, amount of furniture and color (both existing and new).
Make sure he covers everything very well, including furniture and the floor. And, don't pay for a half fast job. It is up to him to do it right and make you happy.
Did he give any references? If so, give them a call. Plus look him up online with the Better Business Bureau.