Assisi to Menton, France or Menton to Vernazza to Assisi

I think it will be extremely difficult to fit all those places into one week. Pienza doesn't have a train station, so you'll need to take a short bus hop to Buonconvento. Total time Pienza to Menton is about 10 hours, so that's going to cost you one full day of the seven you have available. Then Menton back to Vernazza is at least 5 hours, and Vernazza to Siena is about 4 hours--another day gone.

Assisi is lovely, but unless you want to spend an unusual amount of time at the religious sites, 1-1/2 days might be just fine. I visited on a day-trip and decided there was enough there for a non-religious person to justify an overnight stay, but I think 2 or 3 full days might be a lot. Then again, I am not religious.

If I am understanding this correctly you have seven days for Italy and France?
If so it would be crazy to try to see all the places you mentioned. I would suggest visiting your friend in Menton for a few if not all of your seven days using it as a base to see some of the South of France. I have done this twice. Public transportation is very convenient in Menton, and you can take the train over to Italy very easily and quickly, but Assisi is very far away.

Menton is your outlier. Invitations are always nice but it would take up too much of your valuable time traveling to and from. It is a lovely place, as is the whole area, but it's far from your initial plans.