Is saying, "I hope Allah grants me success and I achieve a high score so that I can give charity from it, build a mosque, and benefit the Ummah," considered a vow, and what are the conditions for a vow that obligate one's self?

What you have mentioned is not considered a vow (nadhr), and nothing is binding from it. A vow is only established by a form of words that indicates commitment, such as a person saying: "It is for Allah upon me to give charity," or "I am bound by a vow to fast," and so on. The form of a vow is by saying: "It is for Allah upon me to do such and such," or "I am bound by such and such a vow," or "If Allah cures me, then I am bound to fast for a month," and similar phrases.