Is washing both openings (Istinjaa`) a must before every ablution?
Washing the front and back openings (Istinjaa`) is a must in case they have an impurity, but if they do not, there is no need for that, and Istinjaa` is not a condition for the validity of ablution.
Does an internal examination require ghusl?
Ghusl isn't required for making internal examination; however, it invalidates one's fasting. And Allah Knows Best.
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.
Does touching women temporarily forbidden in marriage nullify ablution?
It is prohibited to touch, or to shake hands with such women as they are considered strangers, and doing so nullifies ablution.