Should a person feel pleased, or have a virtuous vision after offering Istikhara (guidance prayer) in order to do what he/she had offered it for?

The result of the Istikhaarah is not necessarily that a person sees something (in his dream), or feels pleased, but the most important result of the Istikhaarah is whether a person is enabled to do a given matter or not.

After I had prayed, I found a drop of menstrual blood and was unsure when that happened; what is the ruling on my prayer and fast?

All praise is due to Allah, peace and blessing be upon the Messenger of Allah.

Your prayer and fast are correct ,and you should not make up for them, and we ask Allah, The Almighty, to duely reward you for these acts of worship. Your menses start once you see blood because the situation remains as such unless there is certainty or predominant conjecture, as per the Fiqh maxim: "Certainty is not removed by doubt". And Allah Knows Best.

I`m getting old and haven`t got married yet. I love wearing the veil, but people advised me to take it off because it could stand in the way of my marriage, and that I should wear it after getting married, what should I do?

The veil (face drape) is the wear of the Prophet`s (PBUH) wives and righteous female companions (May Allah be pleased with them) and whoever imitates a certain group of people shall receive the same reward or punishment like themselves. Only those who are religiously poor deny women wearing the veil, and thinking that it stops a woman from getting married is from the evil suggestions of the devil.

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.