"...cutter tool similar to a chamfer cutter..." Same tool. It's a bit slow and tedious, but it works. Good thing to do while watching TV.
Regular large rifle primers work just fine. Make sure they're seated properly though. Removing the crimp and cleaning the pocket is usually all you need to do.
Milsurp brass is a bit thicker than commercial brass, so you'll have to reduce the powder charge by 10% and work up.
"...for my 30.06 (Remington 760)?..." Yep. Full length resizing is required any time you use any brass in two rifles. M1's have a tendency to have the extracting brass whack the case mouths too. No big deal. The expander button will fix it. If the dent is too much for the expander button to fit into, needle nosed pliers will take enough of it out to allow the button to fit.