public restrooms

any bar, not usually, yes (unless they are closed that day), they are not set and they are not set

Also, the TI used to have one, right in the back.

Where are public rest room in Lucca

As Nigel said, they're reported to have them at the Tourist Info office. Bars in Italy always have facilities but it's been our experience that they're only for customers so they do expect you make a purchase. A lot of them we've used when stopping for coffee or a beer have had "Customers only" signs on the doors unless they're in a remote part of the building.

But look at it this way: it's a great excuse to rest the feet and do a little people watching while enjoying your beverage! Or just purchase a bottle of water or something to take along.

I'm not sure it's a legal requirement, but I've found restrooms in gelato shops, even small ones. What a great excuse for buying gelato!

Heck, I don't NEED an excuse for gelato!
Nirvana in a cup....

There's also public restrooms on via delle Chiavi D'Oro (nearish the Guinigi Tower) and on via Pescheria (the State Archive should be in the same building, I forget now). Plus there should be restrooms at the station, if you get in by train.