Short answers, pulled from the story.
Search and seizure is the procedure by which police, suspecting a crime has been committed, search a person's property and seize any relevant evidence found. Many countries require law enforcement to obtain a valid warrant or the property owner's voluntary consent before conducting a search.
The Fourth Amendment of the United States Constitution requires that people be secure against unreasonable searches and seizures. A valid warrant is generally required before a search, and most of the detailed law interpreting what counts as unreasonable comes from court rulings, including United States v. Jacobsen and a 2012 Supreme Court clarification on the common-law trespassory test.
The exclusionary rule bars evidence obtained through an illegal search from being used against a defendant at trial. A narrow exception allows evidence to be admitted when officers acted in good faith relying on a warrant they believed was valid at the time but which later proved flawed.
Section 21 of New Zealand's Bill of Rights Act 1990 guarantees the right to be free from unreasonable search or seizure, but Section 4 of the same act denies the Bill of Rights supremacy over other legislation. Courts have found that a search can be lawful and unreasonable at the same time, and individuals who suffer such a search have almost no statutory remedy if no evidence is found.
Article 14 of Italy's constitution limits home inspections, searches, and seizures to cases that comply with measures designed to safeguard personal liberty. Controls for public health, safety, or economic and fiscal purposes are governed by their own specific legislation.
The exigent circumstances exception allows police to search without a warrant when they must act quickly. This applies when evidence is in imminent danger of removal or destruction, when a continuing danger exists, or when people may need immediate assistance. Police in hot pursuit of a fleeing felon may also follow a suspect into a residence and seize any evidence in plain view.