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.
Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.
Should a prayer performer who forgets to offer the middle Tashahod (saying ash-hadu anala ilaha ila-alah) offer Sujood As-Sahw (prostration of forgetfulness)?
He/she should offer two prostrations of forgetfulness at the end of the prayer. And Allah Knows Best.
What is the ruling on swallowing phlegm while fasting?
If a fasting person intentionally swallows phlegm, their fast is invalid, as it is possible to avoid it.
However, if they swallow it unintentionally, their fast remains valid.
Phlegm is impure and should be avoided.