Is it mentioned in the Qur'an or Sunnah that supplication at the Yemeni Corner (Rukn al-Yamani) is answered, and is that specific to the time of Hajj and Umrah or does it include all other times?

Allah has promised to answer supplications absolutely, and He has not restricted this to the Rukn al-Yamani, or to Hajj or Umrah. Moreover, Allah the Exalted is near; He answers the call of the caller. And there is no Muslim who makes a supplication that does not involve sin or breaking family ties, except that Allah grants him one of three things: either He hastens the answer to his supplication, or He stores it for him in the Hereafter, or He averts evil from him.

Among the conditions for the acceptance of supplication are: lawful food and clothing, not being hasty, adhering to Allah's boundaries, and sincerity in action and intention.

Among the times when supplications are answered are: the last third of the night, and between the Adhan and the Iqamah.

As for the Rukn al-Yamani, the Sunnah is to touch it during the circumambulation (Tawaf) and to supplicate between it and the Black Stone with: "Our Lord, grant us good in this world and good in the Hereafter, and protect us from the torment of the Fire." It has been narrated that touching it expiates sins, but it has not been reported that supplication made there is answered.