Incoming tl;dr
Wow, that's really weird, all my experiences with cops have been pretty good. Even when I get pulled over for speeding, they'll generally write a ticket for less than what I was going. Hell, one time I was going 130 down a freeway, and the cop wrote the ticket for 85. I wanna know how you guys get asshole cops, where every one of them I meet is perfectly nice, curt and proper. Even LA cops I've met won't give you a hard time unless you're doing something really bad.
There are also different explanations than "don't have round headlights." It's possible a car matching the description had been involved in an armed robbery near by. Or that a car had been stolen, and they hadn't gotten the plate number for the stolen car yet. Or that a car had been reported that matched the description which had been involved in a drive-by. All this comes to the whole keeping people safe. If you thought these people had been involved in a robbery or a drive-by would you let them walk into the house to possibly get a gun/weapon.
Or it's possible that the cop hadn't met his quota in a few weeks and was facing getting fired. So he was desperate to make a ticket or an arrest. So he could keep feeding his family.
Thinking about things pragmatically and objectively is something I think more people need to learn. There are so many different possible situations that if we merely focus on ourselves and what going to happen or is happening to ourselves. We lose track of the bigger picture, which is that other people have problems that lead them to making mistakes or taking actions as well. If you open your mind to the bigger picture it's not "Oh that fucker gave me a ticket nothing. That fucking utter cock sucking bastard. I don't deserve this fucking ticket." It's more of an "Oh that guy just gave me a ticket, he may have had a reason. Oh well time to pay this..."
Another thing about cops and tickets. Is that once they've written a ticket they HAVE to deliver it. You can't throw away a ticket because IIRC it's considered perjury. And can cost a cop allot of money and possible suspension/firing.
Things to remember about cops, they can't come within the threshold of your home without permission or a reasonable excuse, E.G. seeing a gun, smelling weed/meth or seeing a dirty syringe. They can't come into your home if the object in question is merely suspected(as in 'I think there's drugs or a body in here') to be in the home unless they have a warrant.
Anything you say can be taken as evidence for a warrant, anything you deny can not. So if you refuse to let them into your home, they can't take that as evidence against you to gain a warrant. But if they see drugs or explosives or you tell them about drugs or explosives, they can force their way in with or without a warrant. The garage provided it has a door is considered a threshold. As is the backyard provided it's gated/fenced off.
As soon as you let them in the door either implicitly or explicitly they have full permission to search your house as if they had a warrant. Letting them into the backyard does not count has letting them into the house proper.
They can spot search a car, but can't spot search houses without warrants.
Should your car be in the garage and they have a warrant to search it they may search the car and only the car. If they try to search your garage or house you are perfectly within your rights to throw them out onto their ass's. Just tell them they violated the terms of the warrant. I would suggest you move it out of the garage for them to search it.
If you open the door you implicitly agree to a line of sight search if anything is spotted illegal in that line of sight search, they can come in with or without your permission.
They have the right to arrest/hold you for 24 hours if they consider you a threat to yourself or others. If they consider you a major threat to your self or others after 24 they can put you into a psychiatric ward for up to 60 days. With the possibility to renew it after 50 days should you not be getting better. Should you not be criminal offender this should come out of the states pocket.