Is it a condition that the night vigil (qiyam al-layl) be only prayer, or may it include remembrance (dhikr) and reading the Qur’an, and when does the time for the night vigil begin, and does speaking with someone during it invalidate it?

The conditional vow (nadhr (definition) mu'allaq) is disliked, but it must be fulfilled if its condition is met. If you vowed to spend an entire night in worship (qiyam (definition) layl), and your intention was not restricted to prayer only, then what you did of prayer, remembrance (dhikr), and seeking forgiveness (istighfar) is considered part of spending the night in worship, and you have fulfilled your vow. Recitation (of Quran) is better than prayer during qiyam al-layl if one is able to sustain it; otherwise, one should do what they are capable of.