The restrictions on 12-13 year olds working in Alberta are very stringent, with the restrictions for 14-15yo slightly less so, and so on. It's common for young teenagers to work fast food jobs on weekend mornings, for example, but it would be impossible to earn enough money to support yourself that way.
If you are not a Canadian citizen but on a student visa, there are also strict limitations on work. You will need to have a legal guardian in Canada to get a student visa if you are under 18 - it could be someone on staff at a boarding school, or a relative who lives nearby.
The only boarding high school I'm aware of in Edmonton is Concordia, a small Christian school with both day students and boarders. A more feasible option might be to look for some kind of student exchange where you could live with a family and attend public school in Edmonton for a semester or a year, or just to start with finding some sort of camp program you could attend for a summer, possibly working towards getting a paid position there in future summers.
Drinking age is 18.
You do have the right to enter into contracts for the necessities of life, such as a lease, if you are under 18. However, I think that landlords might also have the right to refuse you. Many hotels have rules about not renting to people under 18, but I do not believe the law requires them to do so. Student residences or youth hostels might work out better.