Allow me to correct #1 to some extent.
In California, a "private person" can arrest someone if he or she saw a "public offense" being committed, or has reason to believe that a felony was committed. Edit: Theft is a public offense.
California Penal Code section 837 (http://www.leginfo.ca.gov/cgi-bin/displaycode?file=833-851.90&group=00001-01000§ion=pen):
The specific section granting a merchant the authority to detain a person whom they have probable cause to believe has unlawfully taken merchandise is 490.5 (f)(1) PC.
or has reason to believe that a felony was committed
A very important point here for a private person arrest acting on a felony is the added element of 'when in fact a felony has been committed.'
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