I suffer from chronic asthma which requires ventoline inhale; which is an air causes the risperatory track/airways to expand. Am I permited to use it during the daylight of Ramadan? And does break one's fast? Noting that I can't go an entire day without using it?

Taking inhaler through the mouth or the nose breaks one's fast, because it is meant to reach the lungs which are part of the coelom. Therefore, whosoever use inhalers for couple of days is obliged to preserve his fasting and make up that day later. As for the ones the ones who need to use it every days are obliged to preserve his fast and feed a poor per day and the cost of feeding one poor is estimated (60) piasters for the year (1421AH). And Allah Knows Best.  

Is it incumbent on the fiancée to obey her fiancé?

When the woman settles in her husband`s house, it is incumbent on him to provide for her and it is incumbent on her to obey him. Before that, and if the marriage contract had been concluded, then she is lawfully his wife and thus she should abide by custom in treating him, but if the marriage contract hadn`t been concluded, then she should treat him as a non-Mahram (Marriageable).

What is the ruling on giving my mother a gift at "Mother's day", is it Halal or Haram?

Praise be to Allah the Lord of the Worlds.

There is nothing wrong with doing that as long as it isn`t done in imitation of the disbelievers. And Allah The Almighty Knows Best.

Is it permissible for one to shake hands with his uncle`s daughter?

No, it isn`t because he is a non-Mahram (Marriageable) to her.