Is it permissible for a Mahram to travel with a woman to a city for an exam and then return? Is there a difference if the city is Makkah or Madinah? Is it permissible for the woman to reside in a house she owns in another city without a Mahram if the Mahram takes her there and brings her back?

It is not permissible for a woman to travel a long distance without a mahram. However, her residing in another country without a mahram is permissible if she is safe from fitna (temptation/strife) and there is no suspicion. If a woman travels with one of her mahrams, there is no objection to her residing in the country where she settles, as long as she is safe. There is no difference between Mecca and Medina or other countries, nor between the dwelling being purchased or rented.