Is the prayer valid for someone who intended to pray alone, then another person joined and became an imam for him? And is the prayer valid for someone who intended to lead the prayer (as an imam), but the follower delayed until he (the imam) finished his prayer?

Changing the intention of a person praying alone (munfarid) to leading the prayer (imamah) is permissible according to many scholars, and this is the most preponderant opinion. It is not a condition for the validity of following an imam (iqtida') that the imam intends to lead the prayer. As for a person praying alone intending to lead the prayer, some jurists have elaborated on this: if he intends to lead while expecting the presence of congregants, his prayer is valid. However, if he does not expect anyone to attend, his prayer is not valid. Some scholars have held that if a person prays alone with adhan and iqamah and intends to lead the prayer, his prayer is invalid. As for a woman, it is not permissible for her to lead men, but her leading women is valid.