Yes they do. I have had to use it twice on AA this year alone for deaths in the family. Both times AA was very accomodating. I called the EXP desk (but any AA number should be able to help) and they have I believe a set price between the 2 airports for the bereavement fare. As long as there is a seat available in a particular fare class you can take the flight. Also, these fares are fully changeable (again as long as the fare class is available on the new flight).

Only 1 in the 4 flights did not have the fare class available to LGA, so I booked to JFK instead. When I got to ORD for the connection, I got on the LGA flight on standby.

Not that is matters when you are traveling for this reason, but you do earn full miles and eligible for any upgrades you would typically get.

They will need information such as person, relationship, hospital, doctor, and phone (of hospital/doctor) I believe. For deaths they need the funeral home name/address/phone, person, relationship.

Sorry to hear about your situation. I hope all turns out ok.