Have you booked your flight yet?
I'd start in Rome and end in A'dam. Rome is hotter early, better to get there as early in the year as possible and work north from there.
I personally wouldn't want 4 days in A'dam. One or two days to see the architecture, canals and a museum or two is plenty. It might be better used elsewhere.
A'dam to Bruges I'd use the train.
Same for Brugges. For me one full day, two nights, is plenty to see the beautiful architecture of the city center.
Bruges to Paris I'd also use the train.
I think you can use any amount of time productively in Paris. The list of sights is nearly endless plus it's a great place to just relax and enjoy a park or sit at a cafe and people watch. I'd probably plan 6 nights for a first visit, just to give yourself more time to do things elsewhere, with a plan to come back again and see things you missed (or just come back).
I went to Burgundy/Alsace/Champagne last May and felt it was too early to be there. The weather was still somewhat cold and the vines weren't really starting to grow.
Train, or fly, to Provence then rent a car. If you have a car no where else, this is the place to plan renting a car. Public transportation is too limiting for the time available. My choice for a base is Arles. There is a ton to see in the area and it should be warm in May. I spent 5 nights in Provence last September and didn't tick off everything on my list. I'd suggest planning 6 nights.
Avignon is a pretty major hub. If you aren't planning more stops, I'd fly to Rome from Avignon, on a European carrier, to save valuable time and money.
Rome is another city with virtually endless sights. If you work through the entire itinerary and have extra days, use them in Paris or Rome.
If you have four extra nights, check out Antibes on the Riviera. My favorite place and four nights gives you time to see some sights (by train, you don't need a car) up and down the coast. If you do this, fly to Rome from nearby Nice. If you absolutely don't want to rent a car. Skip Provence and just fly to Nice and stay in Antibes. I walked from the airport to the train station (at least half a mile), then took the train to Antibes (about 10 to 15 minutes). If walking or baggage is an issue, you can taxi to Antibes.
For lodging, I really like Airbnb in most of France. Just pay close attention to location when you don't have a car. I found a great apartment in Juan les Pins (next to Antibes) for around 50 euro per night in September. Paris is expensive. There are Airbnb's in Paris but look for their tax number to make sure they are doing it legally, Paris is cracking down on others. I've rented apartments from Airbnb for under $100 and found business hotels, like Ibis, for around $100 per night. Either were perfectly fine. It's easy to spend much more for a nicer hotel (or even not so nice hotels).
I haven't used an Airbnb in Italy so I'd make a point to ask others about their experience.
Bear in mind (for some) mentioning Airbnb is an issue (same for Uber). Expect some responses about putting others out of business. I see it more as helping out a person who can use extra euros in their budget. There are "professional landlords", however, who buy multiple flats specifically to rent through Airbnb. You can also rent a room in a family home, which is an interesting way to meet a local and get recommendations/information about the area. I've done both and had good results at least as consistently as hotels (where I've paid more).