Is it possible for police to legally search a car without the owner's permission?

If they have a warrant or probable cause, yes.

Yes. Police don't need your permission to search your vehicle if they have probable cause to believe that the vehicle contains contraband. For example, if a cop pulls you over for a minor traffic violation and he smells marijuana when you roll down your car window, the cop can search your car without your permission. Other valid searched of a vehicle without driver's consent are: 1) "search incident to arrest" and 2) a search done while doing inventory of your car if the police have to tow it.

Generally speaking, No. But there are exceptions.

It really depends on the circumstances, sometimes the police are entitled to a warrantless search and at other times they can search if there is no objection from the person who is currently driving or possessing the car. I would need to details.

Yes. It's called "probable cause."