Tap water is very safe, Croatia has some of the best drinking water in the world.
Croatia has type F sockets, you will need a USA to Euro converter, they should be easily found.
Dubrovnik is expensive. If you did not think it was expensive you must be doing well for yourself. In terms of cooking it depends on where you will stay and if you will have a kitchen to be able to cook. If you are staying in a hotel you may not be able to store cold food. I cannot remember ever staying in an apartment without a kitchen.
Supermarkets have different sizes, there is a Maxi Konzum and a Hyper Tommy on Vukovarska ulica which will have best selections, Konzum locations closer to the old city will still have things to buy but are small.
Agree with Dubrovnik pass. City walls are 40€ on their own, the pass is also 40€ and gets you public transportation while it is active. I usually stay in the old city so that is not very useful to me but it is nice. I can also recommend Lokrum or cable car to Mt Srđ but that is very expensive. I do not know where else you are going so cannot make additional recommendations.
There are not really areas that are not touristy if you are staying on the coast. Especially not in Dubrovnik. They also just changed traffic regulations so be careful, if you use the central city roads without a permit you can be charged.
I have no idea about travel insurance, I usually just take basic insurance that comes with the car. However some car companies will find scratches when you return the car if you do not take their insurance. It is good to photo and video the entire car for damages before you rent. I do this even at car rental companies I have used without problems for years.
Also this may not matter to you but do not get into a taxi. Use Bolt or Uber instead. Taxi drivers may set their own prices outside the apps and some of them are very expensive.