Nice, Nizza in Italian, changed hands between France and Italy a number of times, though it is at heart an Italian town. The last change took place only in 1860 and the people of Nice would probably have counted with the possibility that it would one day revert to Italy again. Nevertheless, the stamp would not be honoured by the French post office and Italy is not really in walking distance of the city centre.
I know of one case where an Italian town is heavily French-oriented: Aosta. To go to France from there is even more difficult, since there are some mountains in the way.
The value of the encased stamps was the denomination of the stamp. In theory, once the scarcity of coins was over, you could break out the stamp and use it without a loss of money. In practice, I suspect that an Italian stamp would pass in Nice, but travelling to Italy to spend it would cost a whole lot more than 5 centesimi (about a halfpenny in those days.)
Peter