How fun! You’ll get many great ideas. I want to comment on #3 and give you one new idea.

Le Marche is tough without a car. We spent 5 nights in Pesaro a couple of years ago and it was a slog via bus to do much of anything other than a train to Ravenna, which was magnificent. I would go there for up to 3 nights. In Umbria it is much easier by train. Consider cute little Spello, just a few miles from Assisi.

Have you been to the Dolomites? Such a different part of Italy. Our favorite place is the Val Gardenia, but I don’t know if dogs are allowed on the buses so you’d have to check. We have long planned to go to Merano which is accessible by train, but haven’t been able to tear ourselves away from the Val G . You might look at Merano and combine it with some time in Bolzano before it gets too hot, or wait until September as that is a lovely month after the summer tourists leave.

You should read the forum article on Italy Reopening. From what I read there and other sources the "reopening" will only be for neighboring countries.

In Umbria, add Spello, Montefalco, and Bevagna. We spent two weeks in Umbria and enjoyed it tremendously except for Spoleto. We also loved our trips to your number 1.
We always rent cars and Umbria and Tuscany benefit from having one to explore.

Bob, we live in Italy as residents, so can travel within Piedmont from Monday onward, and anyplace within Italy without restriction starting on June 3rd.

2- add Vicenza! A little jewel box of a city 😊

You should read the forum article on Italy Reopening. From what I read there and other sources the "reopening" will only be for neighboring countries.

They are legal residents living in Italy,

Being resident you just know that a car could be a plus, mainly to reach some place outside main city.
Anyway:
2) my town of course is reacheable by local train from Bologna: Vignola. Is just starting the cherry season and in a couple of weeks we will be at top of production. And of course there is the fortress, good restaurants and a nice bikeway (or a path) among the cherry-tree orchards where you can easily walk with your dog.
Parma is very nice indeed. Even the hillside offer several towns and villages whom worth a visit. In the plan could be nice the visit to the Masone labyrinth.
Mantova is another gem in the plan. End of June and July are even good months for cantalopes and watermelons.

One of my favourite places in Piedmont is Avigliana, 20 minutes west of Turin on a regional train line through Val di Susa. The Sacra di San Michele is a 1000-year-old pilgrimage church perched on a rocky outcrop above the town. It's on one of the main historical pilgrimage routes (Via Francigena.) There are two little lakes with a wetland park around them, fun for Barley. Great restaurants, though who knows if they will be open. We stayed 3 nights at an absolutely enchanted place, a 500-year-old convent now owned by a Catholic group that runs major anti-Mafia programs. Certosa1515. I don't know if they will be open or if they accept dogs, but the former nun's cells, with private baths, were very cozy and both the grounds and the onsite restaurant were incredible.

As far as the Langhe, we especially enjoyed Neive and Cherasco on our tour through the region. Barolo, Barbaresco, Alba and La Morra are all great, of course.

Cinque Terre: we were lucky to visit in 2016, when tourism was down. Lucky might be the wrong word, as it was due to terrorist incidents like the nightclub shooting in Paris earlier that year. But we were there for four nights and always - always! got the 4 seats facing each other on every local train we jumped on. As far as preferences between the five towns, I'm an outlier for sure in that I preferred Riomaggiore. It had a slightly less-overrun-by-tourists feel and a fabulous rooftop terrace bar up above a restaurant near the train station, where we ended most evenings with a glass of rosé, aperitivi and a sunset. (We also always got a table right by the edge looking out over the sea.) Our apartment there was pretty average but fine. I can send you the info if you want it. Don't know about dog-friendly or not.

oh, more to add.

2) in the Veneto - a little town northeast of Venice, Portogruaro. We had a loft apartment right in the centro storico for ten days. Fabulous town, with little canals, great restaurants, pretty sights, right on the train line from Venice and to all the other main towns in the Veneto (Treviso our fave). Way off beaten path and we really enjoyed it. People were surprised we were there as tourists and that we stayed as long as we did. Only 45 minutes on train to Venice. If I were to go back (ah hem, when I go back), I would add Udine. Also very much want to go to Lake Iseo, especially Lovere, and the Val Camonica for the rock art. (world's first UNESCO heritage site.)

3) Umbria - spent a week each in Todi and in San Gemini. Both are highly recommended, along with the ones you mentioned. Spello and Amelia too.

Besides the above-mentioned towns in Umbria we also enjoyed Gubbio.

Another tip. Honestly, while there are still no tourists (or very few ones) you could visit the most famous cities and monuments. Vatican museums, Venice, Rome, Uffizi in Florence...