Is it permissible to trim the eyebrows if they become excessively long?
Trimming eyebrows is permissible if they become excessively long, yet; it is reprehensible. And Allah Knows Best.
Is it permissible for a woman to uncover her face while performing Umrah (minor Hajj)?
The woman while in a state of Ihram (ritual consecration) is obligated to uncover her face and hands, but at the same time permitted to let her head-covering garment drape from her head down over her face when non-Mahram (i.e., marriageable) men pass by her. And Allah Knows Best.
Should a person who doesn’t offer Tasbeehb (saying Subhaana Rabbiya Al-‘Atheem during Rukoo`, and saying Subhaana Rabbiya Al-A‘laa during Sujood) during Rukoo` and Sujood perform As-Sahw Sujood (prostration of forgetfulness)?
He/ she doesn`t have to perform Sujood As-Sahw, whether he/she didn`t offer Tasbeeh during Rukoo` and Sujood either intentionally ,or unintentionally, provided that the attentiveness of the heart wasn`t undermined since it is a pillar in both.
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.