For addresses where we lived, should we repeat an address? My wife would spend summers in the home country while studying in Canada.
I saw you asked a few questions about address and travel history so I'll explain both here.
Basically they need to know where you were for every day of the last 10 years. So you need to understand the travel history form in the context of your addresses table in the background document.
So for example, if in your addresses you have from 2015-01 to 2016-01 you lived in 123 Street Avenue, Toronto then you don't need to put that in your travel history. Put every other travel other than that. If you went to Paris for a week in 2015-03, then only put that - don't put Toronto from 2015-01 to 2015-03 and then again Toronto from 2015-03 to 2016-01. Because they'll correctly assume that if you weren't traveling then you were at your address. I hope I'm explaining this correctly.
I personally put every trip back home as well because my parents' home is not in my addresses table. So if I went to see my family for the summer in 2016, I put 2016-07 to 2016-08 went to home country for tourism. I looked around the web for this info and a lot of people suggested doing this. Just keep in mind that you are completing your addresses table with the travel record table.
To answer your other question (teaching and studying at the same time), if she was a resident in Canada and has an address for those dates, then you don't need to put that in your travel history. In other cases, you can put anything for purpose of travel. For example, I have a 2 day trip between two other trips and I just put "layover" because I left the airport and stayed one night to get a better price for my other ticket but it's technically a trip because I have entry and exit stamps. But it's not really tourism so I put the real purpose. Since it's not a dropdown, just put whatever best describes the nature of your trip.