Let's assume that the border is open to you by then. (It's fine for your wife and child but, currently, you are not allowed to enter Canada until July 31 and that will likely be extended given what is happening in the States.)
You do not have to leave. Once you have submitted the application - provided you still have status when it's submitted - you can stay until there is a decision.
This the big question that everyone struggles with. If you are coming to get sponsored, you will likely be denied entry. There is something called dual intent, which is the idea that you are coming as a visitor but also applying for permanent residence and if you are denied permanent residence you will leave immediately. I have never done this and don't know what you should say. What I've heard people say is you need to have a main reason for travel - i.e. visiting your in-laws - that is different from getting PR. If you say you are coming to get PR you will (very likely) be denied entry. Of course, lying is an offense under the immigration act and could see your PR or citizenship revoked at a later stage. It's a tightrope. Sorry I can't be more helpful.
Well, if you have a job offer then you can be admitted as a worker and completely avoid question 2. But this assume you'll have a Labour Market Impact Assessment (LMIA) as well - in this situation they issue a work permit at the border. But they won't issue one without the LMIA (unless it's for a job that is exempt: https://www.immigroup.com/news/what-lmia-and-which-occupations-are-exempt )
I'm sorry to say I have no idea. Whichever is more robust/thorough would be my guess.
No your wife doesn't need to have filed her taxes for any American income. That's true for you if you live here and then want to move back to the US, but that's an American rule. (One of only two countries in the world, to the best of my knowledge, that forces non-residents to pay income tax.) Ideally your wife and child need their Canadian passports. If you are driving, they only need citizenship certificates but the government would still prefer they have passports. But, basically, they just need proof of citizenship and they will be admitted. (The passport is proof of citizenship that is accepted more broadly than the certificate.)