My friend and I would like to tour Normandy including the WW2 beaches, Mt St Michelle, Rouen etc. Is it essential to rent a car?
Yes and no. No, you don't need a car if you do an organized tour of Normandy (we took Overlord and thought they were great) and are content to just visit a site or two a day. Rouen and Renne (another great city) are easily reached by rail. Mt St Michelle isn't as easy but you can get there by train and bus/taxi. In Brittany some major cities (Quimper, etc) are also on rail lines. So you can definitely see some of the area.
But many small towns are only reachable by car; technically buses go there but unless you're fluent in French and your travel times happen to line up with work/school hours then the buses may not work so well for you. So scratch Cancale, Dinard, Honfleur, St. Malo, Carnac, Auray, etc. unless you have a car or a lot of time in France.
How do you plan to handle the WW2 sites? There are one-day and two-day small-group (van) tours--most departing from Caen [just noticed this error--nearly all the tours depart from Bayeux]--that are a good way for non-experts to see those sites in the company of a driver/guide who can explain what you are seeing. Otherwise, it's difficult to see very many of the sites without a car of your own. Although I imagine at least a few folks will disagree, I contend that if you don't have detailed knowledge about what happened in Normandy during the invasion, just driving yourself around will not be terribly illuminating.
There are a number of war-related museums scattered around, some of which are accessible via public transportation. The one in Bayeux is newish and a reasonable size. The Caen Memorial museum (a city-bus ride from the center of the city) covers more than the invasion--in fact, it covers more than WWII--and is a full-day affair; it's a fair criticism that it is expensive and also tends to be crowded. At least some of the van tours allow about enough time in St-Mere-Eglise to allow a visit to the Airborne Museum. There are other museums that might be of interest to you.
The nearest train station to Mont-St-Michel is in Pontorson. I understand there's a shuttle bus from there to MSM. But getting from Bayeux (the best place to stay in the invasion area, I think, because it has some additional worthwhile sights, and it has train service) to Pontorson isn't easy or quick by public transportation. The Churchill Hotel in Bayeux runs transportation from Bayeux to MSM once a day and would be the easiest way for someone without a car to get there. However, users of that shuttle will arrive at the same time as many other day-trippers, when the street to the top of MSM is reportedly shoulder-to-shoulder tourists. The consensus is that folks who want a sane experience at MSM should spend a night in that area (difficult without a car) and see MSM either late in the day or early the next morning, outside day-tripping hours.
Like Bayeux and the larger Caen, Rouen has rail service. If traveling from there to Bayeux, you'd change trains in Caen.
If you want to explore auto rental, check in Caen as well as in Bayeux. I think there's just one rental agency in Bayeux; Caen has quite a few, some of them conveniently located very near the train station.
In Normandy I had no trouble visiting Rouen, Caen, Bayeux, Falaise (museum) and several coastal towns as far east as Honfleur without a car. However, I was not pressed for time, and I didn't care about seeing Mont-St-Michel. I also saw a lot of towns in Brittany without a car, but again, I had the time to deal with the limitations of train and bus schedules, because I'm retired. I don't think Dinard and St. Malo are particularly difficult without a car.
I personally consider it essential to rent a car for a trip like this. If you take a D-day tour, the tour guide will drive you to the few sites they visit, but if you want to see anything else, or easily get to MSM or Rouen (+2-1/2 hour drive between the two), you will need a car. This is a very rural and spread out area.