1. Probably better to keep it going but no real risk/benefit. If you earn interest on any deposits you may have to file some tax returns on them. Not sure how that works (see below) but I plan on hiring a cross-border CPA.
2 + 3. Good question, I think a cross-border CPA is best to answer them. From what I understand Canada and the US have tax agreements. Which means while you are a resident of one, you do not owe taxes to the other as long as you are under a certain income. For example, the US waiver I believe is $90k, which means if you earn <$90k in Canada, you owe $0 in US taxes. I'm sure Canada has a similar waiver. I wouldn't worry about owing Canadian taxes until you become a Canadian citizen though (if the income is earned outside Canada). This might be COMPLETELY wrong so don't quote me without checking with a CPA, but I believe only Canadian citizens abroad would incur taxes not PR's. Now keep in mind Canada also has a tax status called Non-Resident for Tax Purposes. My wife who is Canadian filed for that status a few years ago which exempts her from owing any Canadian tax as she is not residing in Canada (not sure if there is a dollar amount like the US but I believe there isn't). Once we go back she will readjust her status accordingly. But I believe this is for Canadian citizens only. I don't believe a PR is required to report foreign income because they are not citizens. Someone please correct me if I'm wrong.