Sworn officers' jurisdiction is restricted to their city or county limitations. Outside of their jurisdictions, the only type of arrest they could make is a "Citizen's Arrest", which by the way does not require a Miranda Warning.
When Detective Curtis takes the envelope from the victim's wife in the park, he does not wear gloves or hold the envelope in such a way as to not get his prints on it. He should have held it the way Detective Briscoe held the wallet, along the edges without his fingertips making direct contact with the flat side of the envelope, where they would dust for prints.