Does assenting to an oath count as an oath, and does saying "yes" to someone who says "by Allah" or "for Allah's sake" count as an oath?

An oath is not constituted merely by assenting to or confirming another's oath, or by saying "yes" to someone who swore an oath. A legitimate oath is constituted by mentioning one of Allah's names or one of His attributes. If the oath is based on truth, there is no objection to assenting to it; but if it is based on falsehood, it is impermissible. The phrase "by Allah, I ask you to do such and such" is not an oath, but rather an appeal by Allah, and the preponderant opinion is that it is obligatory to answer someone who appeals by Allah. Fulfilling the swearer's oath is recommended, not obligatory.