Can anybody use the crossing or do you need a plane ticket to use the bridge?

You need a plane ticket within certain time frame of entry. The bridge is meant only for people who are using the airport. You have to show ticket/boarding pass to cross.

Only with a plane ticket

Plane ticket with, only

Do you need a passport if you have permanent resident card

You can cross the cross border only with the plane ticket that arrives in Tijuana without having or you have to have a flight to San Diego