Is it permissible to repeat a prayer that was shortened, and should it be repeated as a shortened prayer or as four rak'ahs?

Repeating the prayer is not prescribed unless there is a legitimate reason. If a traveler prays a shortened prayer and then takes up residence, he must complete it according to the Shafi'i and Hanbali schools. It is more prudent to repeat the entire prayer to absolve oneself and to avoid disagreement.