No. You don't need to pay this since he wasn't removed. You just need to pay the TRP fee. Be aware that it's not uncomfortable for TRP processing to take a couple of years.
You should leave within 30 days. You have to stop working immediately and your children are no longer authorized to study, effective immediately.
If you want to try to stay, that would involve either trying to appeal the decision or applying for restoration. On what basis did you apply for the...
No, IRCC will not expedite PR for this reason (or really any other reason).
What is stopping you from traveling for a visit to your home countr? There should be no reason why you can't do that as long as you have a valid eTA or TRV to return.
I'm also very sorry for your loss.
That's not really accurate regarding H&C being a separate application. You can apply for PR directly through H&C, that is true. However you can also request H&C consideration as part of an existing economic or family class application...
You need to wait until you have AIP before you can apply for the open work permit. She doesn't meet the requirements to apply with AOR since she is out of status.
IMO you have to find a path somehow through the existing rules. Any announcements IRCC makes for next year almost certainly won't help you. Or make preparations to leave.
You should be prepared that the application is going to be refused since you were clearly common law at the time you landed and became a PR but failed to declare your partner. Your partner will need to quaify and apply for PR on their own through an economic immigration stream like express...