No, they cannot/will not check your Canadian driving record, but you can easily have the provincial ministery of transport print that for you from whichever province you most recently had a license in. In Ontario it costs only $12 for a "driver abstract". You can even get them on-line.
No, you should not have to pay duties or
taxes on your personal goods.
U-Haul is the only truck rental company that allows one-way cross border moves. Although you might also check with Penske, I think they *might* do cross border now, not sure.
If you want a proper "moving company" several do cross border moves, but you will have to go ahead of your stuff, and then send the moving company a copy of your approved entry (photocopy of your visa stamp from the POE) otherwise they can't move your stuff in.
You could also try UPak, a moving company where you pack your stuff into a container and they pick it up and take to where you're moving and you unpack. Again, you will have to go ahead of your stuff, and then send them a photocopy of your approved entry.