1

It depends on each specific place as Patagonia is a huge region about 2000 kms north to south and 1000 kms west to east. Patagonia is a paradise for camping. There are many camping grounds with services, others without and for free and at many places you can do also wild camping.
At many places in Patagonia it is pretty safe to leave most things just in the tent while out for trekking. At some you will have lockers. Of course don´t leave money, valuables or electronic devices. If you camp inside or close to bigger towns you should be more careful like Bariloche, Ushuaia, Punta Arenas, Puerto Natales or Cohaique but even there it is very unlikely to have any problem. Being in Patagonia you should experience some multidayhikes and camp out in the wild. Around Ushuaia along the Paso de la Oveja or along the coast or the Carabajal valley this is a great option. Also in el Chalten it is well worth to camp at Laguna Torre as you will have a stunning view of Cerro Torre early in the morning at sunrise. For the rest of the day it is in clouds most of the time

2

The early morning (and late evening) view at Laguna torre alone is worth taking a tent to patagonia!

3

In Argentina, camping is the cheapest option and there are campsites at all towns and villages.
In Chile, camping is only interesting in the national parks and in the nature. In towns and villages, it is better to stay in family-run accomodation.