Do they try to find the person who got your card and was using it?
Yes. And depending on the circumstances they're almost always successful.
Can someone with bad credit co-sign with someone with good credit and get a card using the other person's good credit?
You've got that stated backward. The person with bad credit who needs a credit card would be the borrower. He has to find someone with good credit to be the cosigner. And the cosigner has to really trust the borrower because he is going to be responsible for repaying the card if (when) the borrower doesn't.