Both can be done.

First one is more common. Second one depends on your industry and who your competitor is (sleaziness)

The fighting is most fierce near communization, those edge cases where a brand name is becoming a common name for the item (think Kleenex).

I suspect it's not actually as bad as people may think (if someone searches "Dominos" are they looking for any pizza place, or that particular one), but it does happen.

Try to drum up public outrage to get them to stop? Maybe if they use your trademark there's cause to sue? Otherwise good luck.

Branded search terms cost way more for competitors for reasons I can't be bothered getting into, most likely they are losing money by doing it. You can counter it by bidding yourself - which on branded search terms is usually dirt cheap. But ultimately, the only real counter is delivering a better service.