It depends upon the school.

Some schools may accept your COPR/Record of Landing as proof of residency but some will likely ask you for a PR Card.

The good news is that, if you get the request for a PR Card in writing from the school, you can submit a copy of that request to IRCC and ask for urgent processing of your PR Card.

Drive.

You are not required to present at PR Card at a land border as long as you are in a private vehicle (i.e. your car). You will need to provide your COPR/Record of Landing, though.

No, when you swear your Oath of Citizenship you will cease to be a Canadian Permanent Resident and are supposed to relinquish your PR Card at this time. (This is before you have a Canadian passport.)

This should be sufficient. IF IRCC wants more information they will let you know.

They should be. However, in the early days of the eTA system (that’s the thing you have to get instead of a visa) many permanent residents were able to get eTAs and come to Canada, which they weren’t supposed to do. That’s changed but we don’t know how thorough the system is. The only way to find out is to apply for your eTA and see if it’s approved or not.

If you didn’t get back before the pandemic, you are probably going to have problems keeping your status.

Under normal circumstances the best thing to do in this situation is to fly to the States and have someone come down and pick you up and drive you into Canada. (You can enter Canada by private vehicle without a PR Card.) There is still some risk of your status being revoked in this situation.

But, honestly, we don’t know how the pandemic is going to affect Canadian PRs living outside of Canada who were not meeting the residence requirement. There’s probaly very little political will for an amnesty in this situation and so it’s definitely possible that there will be no official extension/waiver of the residence requirement in cases like yours.

You need to get her a legal name change from your province of residence.

If her only ID is her passport she will have to have her name changed on it as well. And so it depends upon the country, but some of them will likely not accept a Canadian name change as proof of a name change. So that’s something to think about.

Once that’s done, you will have to apply to change her name on her Confirmation of Permanent Residence (COPR). Once that’s changed, she can get a new PR card with the new name on it.

Processing time depends upon many factors, including whether or not the person processing the application believes the applicant has met the residence requirement. If the government suspects your son hasn’t met the residence requirement, it will take a lot longer to process the application.

There is not such special permission to travel while waiting for a new PR card. What your nephew needs to do is contact IRCC to request urgent processing due to a family emergency. He will have to prove it’s an emergency.

Let’s try to answer your questions:

1. There is some small chance that your PR Card could be confiscated when you try to enter or even that your status could be revoked, but this is not likely to the best of our knowledge. More than likely they will let you in but when you apply for new PR Cards you will be issued 1-year cards rather than 5-year. They may also not ask you many questions and you might be fine. It depends upon the officer you get. Whatever happens you will not be admitted and then immediately asked to leave.
2. You will not be admitted and then given a departure order.
3. You should both try entering first before worrying about this. But, if they are threatening to not let you in and revoke your status, you can explain that it wasn’t your son’s choice and he should still be allowed to enter with valid status. If that doesn’t work, he can make a “Humanitarian and Compassionate Grounds” appeal to keep them from revoking his status.