I used to live in Rapid city, and recently moved to Minneapolis. My wife went to India, before i moved. Her first port of entry is MSP. Can she get the baggage at MSP, and exit the airport without boarding the last leg of her flight i.e. MSP to RAP. Should she tell the airport staff in the first leg to tag the baggage only till MSP? Or can she collect the baggae after custom check and exit?
No, the airline will generally not allow "short checking" bags to a point that's not your final destination.
However, if MSP is your wife's port of entry into the US, she has to collect her baggage for the Customs check, so there's nothing stopping her from exiting the airport and taking it with her. Airlines are not fans of people not showing up for their connecting flights, because this is often used for a cheap flight tactic called "hidden city ticketing" that is against the T&Cs, but for a one-off there will be no consequences.