1) First of all you have to get your car into the US. If it's leased, this is pretty much impossible because the lease company won't release the title to you. If you've owned the car for at least a year there is no duty or tax whatsoever regardless of what the CBP officer might think, it's classed as personal effects. However, you need to check the DOT regs to make sure it's legal in the US (you might have to make minor alterations to it) and also you must have the EPA test done at a local test centre (and by "local" that usually means the nearest place by a port where cars are usually imported in bulk, although some garages can do it, but not many). Once you've got the CBP release/EPA paperwork you can transfer the title and get a new registration.
2) Read this:
http://www.cra-arc.gc.ca/E/pub/tg/p151/README.html
If you're on a TN-1 it is generally a bad idea to move your tax home to the US for two reasons, the first is that TN-1 requires residential ties to Canada (as it's a temporary worker permit), and moving your tax home to the US means you have to cut all residential ties to avoid dual taxation, so they work against each other.
The second reason why is if you ever get refused entry you automatically become resident in Canada again, which means a ton of tax paperwork.
So you carry on filling out a T1 for wherever you lived in Canada, and file a 1040NR and 8840 or 8833 in the US to claim the tax treaty exemption. You file a copy of your W-2 with your T1 and claim the foreign tax credit in Canada to avoid paying
taxes twice. The general guide for the T1 includes instructions on how to do this.
3) TN-1 is not dual-status, it confers no elgibility for LPR applications. Your employer can sponsor you for LPR status (provided you meet the qualifications, which are tough), but it is tricky if you're on a TN-1.
4) LPR status is not citizenship. After you've been resident in the US as an LPR for five years then you can apply for US citizenship, you can keep your Canadian citizenship and be a dual citizen.
Bear in mind however the obligations of being a US citizen, as soon as you become an LPR you must move your tax home to the US which means you must sever all residential ties to Canada in order to avoid dual taxation. LPRs and US citizens must file a US tax return every year during their lives unless they cease to be LPRs or renounce their citizenship, regardless of where they reside.
The US foreign tax credit maxes out at $80,000 and something (at the moment, they're talking about repealing the limit), so if you became a US citizen and then moved back to Canada, you would be subject to dual taxation on earnings over that limit.
Steve.