Can the police search my house without a warrant?

The automobile rules don't carry over. A home is the most protected place under the Fourth Amendment, and the default is that police need a warrant.

The exceptions are narrow and specific. Consent from someone with authority to give it. Exigent circumstances — an emergency, a person in danger, evidence being destroyed, or hot pursuit. A protective sweep incident to a lawful arrest, limited to looking for people rather than evidence.

What is *not* an exception: an officer's belief that a warrant would have been granted, the seriousness of the suspected offense, or the fact that a door was open.

Exigency is where most of the litigation happens, because it's asserted after the fact. Whether the emergency was real, whether it was created by the officers' own conduct, and whether there was time to get a warrant are all examinable — and telephonic warrants have made the "no time" argument harder to sustain.