I might be reading it wrong but your wife will need TD Visa from the consulate. She can't get it from the border. As for your son, I'm not sure if consulate will entertain his application if he's a Canadian too. This said, if you got TD for your son at the border on his Canadian passport, can he...
Btw, just ooc, if you had your citizenship certificate (assuming you naturalized) or your birth certificate (if you are Canadian by birth), wouldn't that be enough to at least enter Canada? This way, one wouldn't have to worry about temp passport and simply get a new 10 yr passport using urgent...
1. Your degree and job offered defines the TN occupation
2 and 3. You cannot just go to the airport to get the status (not Visa for Canadians). You have to be requesting entry into the US and have to take your flight
I9 is verification by your employer that you're working there legally and that a designated authority has validated the authenticity of your documents. Regardless of whether you get tn through CBP or USCIS , it's a required step in your employment
Degree: evaluated to US standards if not from Canada or US
Passports and birth certificates for kids and marriage certificate if applicable if you plan to get TD for them
You don't have to carry docs every time you cross the border