Although my dishwasher has a "pots & pans" setting, I've heard it's
not a good idea to wash them in the machine. If not, why not? Are
there certain kinds of pots and pans that are dishwasher-safe and
others that are not? Copper? Aluminum? Teflon coated? I assume that
those with plastic handles should go in the top rack, right?

I throw everything in the dishwasher except pans and delicate glasses.

Very occasionally things break but I'd probably break more if I did them by hand.

Get scientific
- use a Darwinian approach

If they don't survive - bin 'em

For the Female, dishwashers release them from mundane tasks.
For the Male, dishwashers release females for much more interesting tasks.