Siena

Have you checked any of the recommendations in Steves' Italy guidebook for Siena? He lists maybe 20 hotels of various descriptions. Balconies are hard to come by but bathrooms are always included at all hotels. Most exclude pets but you need to check specifically with each hotel if that is critical. If a site is not crowded then it probably not a very interesting site. The locals go to restaurants, churches, but don't go to tourists sites because they are seen them many times before. The locals walk the streets, go to dental and doctors offices, frequent the local bars and the open markets, and, in general, do the same things that you do in your home town. If there is a soccer game in town, you will find a lot of locals at the stadio cheering for the local team. They don't frequent the local museums unless it is a free day. Then it is super crowded. Lots to see and do in Siena. You might want to review some good guidebooks and travel DVDs at your local library.

I stayed at the Albergo Alma Domus - my room had a bathroom, and a phenomenal view of the Cathedral, etc. The balcony was too narrow to sit out on, just stand. Some of the rooms have that view, some don’t, so you’d have to ask. You’d also have to ask about pets, but I certainly didn’t see any.

The premier site for hotels is booking.com, with reviews by actual guests.

Thanks to everyone who responded .

I'll second Alma Domus. Former convent with great views and great price. Smallish balconies, but has one just the same. Plus it is far enough off the more beaten path that you have quiet nights. Siena has tourists like any other popular town in Italy. Less crowded than Florence, but the evenings can get crowded in piazza. For a more local flavor, try the areas around the two parks (giardini) to the west. Lizza has a nice play area for children and the other has great views from the ramparts.

Also, it is my experience that the locals do like to hang out in Piazza del Campo as much as we tourists do. It is, after all, their town.

One thing I found by accident was the museum of the Onda Contrada, where Via Giovanni Dupré, Via di Sant’ Agata, and Via di Fontanella meet. The guide was a member of the Onda Contrada, and did a great job explaining the Contradas, Il Palio, and how they fit in with the life of the city. I gather that at least some of the other Contradas have their own museums also.

Per the advice of someone else on the forum we booked this place for our own trip last May: http://www.laterrazzasulcampo.com/en/
It's a beautiful place with spacious suites, an excellent catered breakfast served in the room, and at least a couple of the rooms have balconies overlooking the Campo - was a great place to have a coffee and watch the city wake up in the morning.
The best strategy to avoid the worst of the crowds is to get an early start before the day trippers start to arrive, or wait until they've departed later in the afternoon before venturing out to the most popular attractions. We found that most of lesser known attractions were hardly ever crowded since the tour groups tend to congregate around the majors.
Siena without the crowds (ie early morning and late afternoon/evening) is a whole different experience than at midday.