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.
What is the ruling on using eye drops while fasting?
Eye drops do not break the fast, even if the taste reaches the throat, because the eye is not an open passage to the body cavity.
Does using a respiratory device (oxygen) affect the validity of fasting?
Using an oxygen device does not affect the validity of fasting because oxygen is a gas with no physical substance.
However, if medicinal substances with physical particles are added to the oxygen, its use would break the fast, as it enters the body cavity through a natural passage.
What is the ruling on the ablution of one who cuts his nails, and is it permissible to cut them before the ritual bath for major impurity (janabah)?
Cutting nails does not invalidate ablution, and it is permissible to cut them before the ritual bath for major impurity. And Allah the Almighty knows best.