do the towns really close for lunch every day?

Towns don't close but businesses do. More places close in towns and villages than in cities.

Regarding the third part of your questions - yes, more than once I have seen virtually all the restaurants and cafes in a town close so that their staff can have the Pausa. Often, some places stay open, especially bars. I don't recall ever seeing a gelateria closed in the middle of the day.

If you are in a touristy town it shouldn't be too much of a problem but if you are in smaller places it can be. Just plan on a gap in the middle of the day where you won't be able to do much besides seeing the town from the outside. Churches don't tend to be locked so you can see these and public spaces like parks are also open of course. When we travel in places like this we often just give up and do as the locals do. Have lunch on the early side to make sure you get to eat and then take a break in the early afternoon. We have found that in most countries that have this practice the whole day is just shifted forward. When things open back up they stay open later and the population is also out and about later, especially in the summer when it can be very hot in the middle of the day. It's really great to be in the main piazza of an Italian town at 11PM in the summer when everyone is outside relaxing, cooling off, and enjoying life!

My only experience in Italy was in the Ortler Alps, but yes, in small towns, everything except restaurants closed for a few hours during the middle of the day. Even the ski gondolas- although this was in the summer, I can't imagine they would shut down mid-day in the winter.

The other surprise was that restaurants don't serve dinner until after 19.00.

Only the smallest towns would close completely - at least the bars will be open. If you are traveling from one small town to another, those pausa hours (1-4ish) might be a good time to travel. Or find a cafe and enjoy the pausa like the locals do.

In Tuscany and Umbria, NO, but in Puglia....closed, Chuiso, Shut from noon -3pm. Better plan to eat before that or after that, cuz nothing is open. Try doing a bike tour and arriving in the next town, half way to your night's destination, and no restaurant, bar, cafe is open serving food! We managed a couple of times going to a grocery that was open and bought paninis at the deli/meat counter.

If you are concerned, stop early in the morning at a cafe/bar and see if they have some sandwiches to pack in the car with you...or go grocery shopping....much cheaper, fun to do, and more variety...buy some water/flavored water, a box of cookies and bread and meat...and you are set.

When we were in Tuscany we found many places closed from noon to 2:00. The restaurants were open.

Restaurants and bars stay open, obviously. Unless it's a restaurant that opens only for dinner. It would not be wise for a restaurant to stay closed at lunch. Whoever says restaurants close at lunch and you have to eat before noon does not know what they are talking about. Italians eat lunch at about 1pm to 2pm. No Italian would go to a restaurant at 11 to have lunch.
What closes at lunch are shops. Opening hours vary from region to region and municipality to municipality but generally the lunch break is from 12:30 to 15:30 in winter and 13:00 to 16:00 in summer. Large supermarkets chains, such as Coop and Esselunga, very common in Tuscany, observe an "orario continuato" and don't close at lunch. These large chains are generally outside of historical centers, but if you have a car to visit those towns, you'll be able to reach them. Non food department stores chains, such as Rinascente or Coin also stay open. There are exceptions to closing times. The historical centers or major tourist localities, have no mandated opening/closing hours, so they do what they want. Some shops may close others may not, depending on whether it makes business sense for them.
Offices and banks also observe a break and close. But I don't think you need to deal with them. ATM are always working.
I don't think you have to worry about about getting bored.
And yes, dinner is not served before 7 or 7:30pm. Most restaurants start opening for dinner at that time.

After three trips to Spain and multiple trips to Italy, I can say that Italy is much more liberal with their hours of operation than Spain! I have NEVER had a problem finding a restaurant open for lunch in Italy. Not so much in Spain. Roberto is (of course) correct that Italians usually lunch @1PM. Shops will close, and so your souvenir shopping will come to a screeching halt! This is a good time to take a mid-day meal, a glass of wine, and a rest. Everything will open by the time you've finished your lovely lunch! In Spain, I found businesses and restaurants to close from 1Pm - 4PM, and it was a total pain!

Again, if you will be in a larger city such as Rome, Venice, Florence, etc., probably almost nothing will close.

FYI, it's called "Riposo".

We found many restaurants closed between 1/1:30- 3:30. This was not the case in Rome and Florence but along the way in Volterra, Montepulciano and Orvieto. A restaurant or two was open but almost NO shops were open. Also, some churches were closed. Notably the one in Rome with the St Peter in chains.

Laughing at myself. Thanks, Rik, the pausa is what buses do.

As usual, Roberto's information can be relied on; he's Italian, of course. What he didn't address is some of the subleties that may confound a non-Italian tourist. Many small shops don't have posted hours. They may re-open at 3ish, or 4ish, or . . . Even if a time is posted, you should treat it as a rough estimate. It's a good example of the observation often made of Italians, that they work to live, not live to work. If there's somewhere you want to shop, do so before the riposo. Churches, museums, and other tourist attractions will probably have posted opening times and adhere to them. Also, our experience has been that many restaurants don't open for dinner before 7:30 or 8. We've taken to eating our main meal midday, around 1:30. Since there may be nothing else to do, it's a perfect time to enjoy a slow, relaxed meal.

I'll augment Roberto's reply (which, as usual, I agree with 100%) with the following:

The further south you go, the longer the midday closing. In Milan, it's 1-2 hours; in Rome, small shops can close for 3 hours (1-4 PM).

The smaller the town, the fewer places will be open during the midday closing.

The more touristy a city, the more flexibility in the hours. I haven't been to Florence in many years, but I remember that some restaurants opened for dinner at 5 PM (unthinkable for any place catering to Italians, so beware of quality in such places).

Restaurants often only have one seating per table per meal. So, even if a restaurant is open from 1 PM to 3:30 PM for lunch, you can't just turn up at 2:30 and expect a table.

As for what to do during the closing, research options that stay open during that time. But yes, if you want to focus on smaller towns, you should indeed plan an early start, a midday meal and rest, then resuming sightseeing in the later afternoon. If you try to "fight" this, you'll lose and be miserable.

Edit: cross posting with Rosalyn, and I also agree with her points, particularly about the looseness of hours in Italy.

If you are driving, then you can explore a town in the morning and drive further in the afternoon. If you're going to picnic, then do your shopping before shops close.