What is the ruling on looking at and shaking hands with female relatives, such as the paternal aunt's daughter, and what are the limits of looking at the married daughters of maternal or paternal uncles who are older than the questioner?

Your father's paternal cousin's daughter is not a mahram to you. It is not permissible for a man to shake hands with a non-mahram woman, unless she is an old woman who is not desired. Scholars have differed regarding the permissibility of looking at the face of a non-mahram woman without desire and with assurance against fitna (temptation). Some, like Ibn Aqil and Al-Shafi'i, permitted it, and this is what people, especially neighbors and non-mahram relatives, cannot do without. As for an old woman who is not desired, looking at her is easier.