What are the rulings on repeated vows, and is expiation obligatory for a vow that was not fulfilled or a promise that was broken, when there is a vow that is not fully remembered?

Your vow to fast on Mondays and Thursdays is valid and binding. You must make up for the days you missed. As for expiation (kaffarah), it is obligatory according to the Hanbalis for failing to fulfill a vow without a valid excuse. However, if you broke your fast with an excuse, there are two opinions, and the preponderant one is that expiation is not required. If the "pledge" (al-ahd) was uttered verbally, then its ruling is that of a vow, and expiation is required if you fail to fulfill it; otherwise, nothing is incumbent upon you. You are not required to offer expiation for a vow you are doubtful about. If you are certain about the vow but doubtful about its fulfillment, then fulfillment is incumbent upon you.