Is it permissible for the young woman to turn the suitor down if she had no feelings for him?
Yes, it is permissible for her to do so.
Is it permissible to swear by (Allah "The Living")?
Yes, it is permissible because "The Living" is one of Allah`s Attributes.
What is the ruling on reciting verses of the Holy Quran on water, then drinking it?
It is permissible to recite Quran on water, then drink it as a kind of treatment since the Quran heals. Allah, The Almighty, Says (What means): "We send down (stage by stage) in the Qur’an that which is a healing and a mercy to those who believe: to the unjust it causes nothing but loss after loss." [Al-Isra`/82].
Is washing the private part after urinating (Istinjaa`) a condition, and is it done with water and soap, or with water only?
Istinjaa` is obligatory for removing impurity, and it can be done with toilet paper, or a stone, or water. It is preferable to do Istinjaa` first with paper, or stone, or the like, then to wash the private part with water until making sure that the impurity has been removed. Using soap is not a condition, but there is no harm in doing so provided that it is washed off with water. It is also permissible to use either water, or paper if impurity is removed by any.