SMU is a respectible school, I don't know the history of how it was named. I've never heard of it being Catholic, any more than St John's, Saint John, St. Catherines, etc are. It is close to Dalhousie as well, lots of university students in that area. SMU is also home to the Sobey's Business College, (or shcool of business or something). It is also home to the Huskies, ( there is one in every province;-)).
Some things you should know, it does get pretty wild with students downtown, the Dome is a popular spot and almost every bar has $1 shot nights, though the city is trying to put the brakes on that. If you're coming to study or party you can do each, but I don't suggest you try doing both. It is easy to fall into the party crowd and blow all your savings, and tuition.
There are certain areas to avoid at certain times, but my daughter has gone to Dal for a couple of years and never encountered any misfortune during her commute. If you are a night owl you don't have to go looking for trouble, it'll find you, but its like that in any city.
Things like bus passes should come included in your student fees, they do at Dal. Check that out, and once you get the bus system figured out it is quite easy to get around. There are also a lot of discounts that students can get from various merchants.
I don't know if you are planning to live on campus or sharing accomodation with roommates. Rent and living costs can vary quite a bit depending on which option you choose. If I think of anything else I'll post it.