If you give them permission to do so they can, or if they have reasonable suspicion to suspect something is going on. If you are being charged with something else (search incident to arrest), they can also search your automobile. Bottom line is that driving is a privilege and with privileges comes a certain level of responsibility. However, your responsibility as a citizen trumps your responsibility as a motorist when in the situation of being pulled over. You have the right to remain silent, refuse consent to search, and the right to an officers name and badge number.
Yes
Yes they can search it
yes, it is not considered a "search"
no
If they have probable cause.
I think they only can search the car if they have probable cause.
Police are liable for the damage that they do to your car. You can file for damages against the police department to get your money.
I think that would be rather difficult for you to search your car while in handcuffs. Now, can the police search your car while you are handcuffed? Sure if they have your permission or if they have a search warrant.
Prowl is what they do, not what they are. To go about stealthily in search of prey
The vehicle can be searched with probable cause, or if the car is being impounded an inventory will be done. The police do sometimes ask for permission as well, and that is another way the search can be valid.
The fourth amendment protects citizens from an unreasonable search of your car.
NO they can not unless there has been a warrant to do so.