Is it permissible to give money in lieu of fasting a full month that was vowed after a certain event occurred, if one is unable to fast it? And what is the amount of this money? And what is the ruling on someone who made several vows and then forgot them?

Fulfilling a vow is a confirmed obligation. Whoever vows an act of obedience must fulfill it as long as they are able. If they are unable, they must offer an expiation for breaking an oath (kaffarat yamin). If you are capable of fasting or hope to be able to in the future, it is not permissible to deviate from it. However, if you are continuously unable to fast, you may resort to the expiation for breaking an oath.

Whoever made several vows and forgot them must estimate them based on what they most likely recall, adding a bit more for precaution. Then, they should fulfill them if they are able. Otherwise, they must offer expiation for each vow with the expiation for breaking an oath: feeding ten poor people, or clothing them, or freeing a slave. If they cannot find any of these, then fasting for three days.