Is it permissible to perform Tayammum for a month due to a medical condition that requires avoiding water?
Tayammum is permissible so long as the above excuse lasts. And Allah Knows Best.
What is the ruling on having slight discharges of menstrual blood during daytime of Ramadan, and before Maghrib time?
A woman who is certain that her menses have started during daytime of Ramadan is in a state of menstruation, and her fasting is considered broken at first sight of blood. However, she is rewarded for not breaking fast at the beginning of daytime.
Is it permissible for someone who has forgotten to mention the supplication pertaining to answering the call of nature to say it while in the toilet?
It is disliked for a Muslim to mention Allah while in the toilet, but he/she may recall the supplication in his/her head without uttering it; that is in case he/she had forgotten mentioning it before entering the toilet.
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.