My questions:
1) What strategies should I use in applying for the spousal sponsorship for my future wife? Can I start the process before getting married? (Ideally I'd like to bring her back to Canada shortly after marrying)
2) It sounds like doing an outland application would be best, but I want to figure out a way to remain with her after we're married, so how do I do that if she can't get a Canadian tourist visa, and I can only stay in Brazil for a maximum of 180 days?
3) Should we use her maiden name for the PR application?... or should I have her get a new passport using her new last name and then use her new name for everything?
You cannot submit the application before you get married, but you can use this time to fill in the forms and gather evidence that your relationship is genuine. In particular, look at the evidence you have now and see if it is lacking - you're lucky in one way because you've got lots of time to remedy the deficiencies in your evidence. Keep phone bills - a phone her some way that keeps a record, even if it costs more. You can get records of Skype and instant messaging calls, btw. Keep all evidence of trips to see her - plane tickets, boarding passes, hotel and restaurant receipts, and lots of photos. Get evidence that shows you and her children have a relationship.
Staying with her on your 180-day visa is one option. See if she can get a TRV to visit Canada - she will need proof she will leave Canada at the end of it, such as a house in Brazil, a job, family, etc. (Dual intent is in the regulations, but few have had any luck getting the visa officer to apply it.)
It would most likely be quicker to use her maiden name for the PR application. Her getting a new passport and new ID with her married name may take a while after the wedding, delaying things a bit. She can always change her name after she gets the PR visa and lands.