All good - thanks for the detailed answers.
It is rare to find somebody going to Calais so informed.
It sounds like you are well informed and have done good planning.
I, like the poster above, am cringing a bit at just 10 weeks to get an Italian visa. I hope it goes very smoothly for you.
A16 Calais - or nearby your B&B - to A25 at Grande-Synthe just before Dunkerque (careful, very sharp corner, hidden cameras all along the A16, speed limits up and down) to Lille (avoid rush hour), Calais to Grande-Synthe also known as E40, from there for a long way also known as E42. From Lille A27 to the border, then in Belgium the road continues as E42/A8 and onto E42/A16 north of Tournai and then the E42/A7 all the way through Mons to E42/A15 over Charleroi to Namur where you turn south onto the E411/A4 which changes to E25/A4 between Bastogne and the Luxembourg border. At Arlon on the border it becomes the A6.
Now from Luxembourg (cheapest fuel in central and western Europe, state set prices the same plus or minus a couple of tenths of a cent per litre, even at motorway services or mom and pop filling stations or anywhere else, and all brands) you can stop or visit Luxembourg, Trier (next door in Germany) or Metz (just south in France). Or continue various routes to Schengen (where the visa is named for) picking up the A8 as you cross into Germany to Saarlouis and A620 to Saarbrücken then signs down to free bit of the French A4 to Strasbourg. Or from Trier, onto the A1 as it winds through the forests to Saarbrücken and to the A4. Better to follow GPS in this area and following signs than trying to follow road numbers in the Saarlouis and Saarbrücken area.
Then the E25/A4 into Strasbourg. Then either E25/A35 to St Louis just at Basle/Basel. Or follow the E52 over the Rhine at Strasbourg via Kehl in Germany to the (in)famous A5 which runs (yes, runs) all the way south to Basel. You could visit Baden Baden or stop into the Black Forest (Gengenbach is close, a great stop and close to Strasbourg).
There are many ways to skin a cat going through Switzerland, including Luzern, or the Berner Oberland, including putting your car on a train under the Alps from Kandersteg - or a long car tunnel called the Gotthard. Your choice, when you pick a route we can chip in further.
Switzerland driving - slow DOWN in tunnels, don't exceed the usually 80 kph limit, both for safety and saving your pocket fine money; and that Vignette mentioned earlier.
One final thing for the moment - you say that you are renting a car for this road trip but then you say that you won't have a car when settled??? Just curious - if you are dropping this car in Italy after you and pooch move, are you aware of the huge drop charges for renting in one country and returning in another? Or have we got the wrong end of the stick? Also, do you know that for the French and Italian parts of the drive all drivers require an IDP?