The most basic/common-sense thing to do would be to test the voltage in the light sockets that are burning out bulbs while the AC is running. That should be your first step.
Even small air-conditioners like I have in a window in my house tell the consumer that they must be on a dedicated circuit.
I am an IBEW Union electrician and have been messing with electricity for half a century. It is a sad fact but contractors are the same as auto-mechanics, they are there for the money and a lot of them are pretty bad, they will screw you any way they can, and do anything they can to keep from taking the blame for anything. If you do not know how to wire a building or work on a car, then you are going to get screwed by both electricians and auto-mechanics the rest of your life. I feel sorry for most consumers that have no mechanical/electrical experience, because I know they are going to spend their lives paying big-bucks for a lot of things......
I try to get people to read books and tackle their own projects all the time, but most of them simply do not want to even give it a try. Almost anyone of normal intelligence and middle-class or lower is better off doing anything themselves than trusting someone they do not know to do it, even if they have never done it before. Sad fact.
I also have worked in a heavy metal fab shop, went to school for tool&die, lived at a motorcycle dealership and have worked as a mechanic and always have done all my vehicle repairs. I was apprenticed to an old-world watchmaker. I have seen so many of my friends and acquaintances get royally screwed it is hard to believe. Sorry.