What is the semi-circular structure attached to the Holy Kabah?
The semi-circular structure attached to the Holy Kabah is Hijr Al-Kabah (is a low wall originally part of the Kabah) and passing behind it is an obligation while making Tawaf because it is a part of the Holy Kabah. And Allah Knows Best.
Which takes precedence: the 'aqīqah or the uḍḥiyyah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The 'aqīqah is an act of worship through which a servant draws closer to Allah the Almighty in gratitude for the blessing of a newborn child. It is a confirmed Sunnah for those who are financially capable, and Allah does not burden any soul beyond what it can bear.
However, the uḍḥiyyah takes precedence, as it is a confirmed Sunnah established from the Prophet ﷺ through stronger and more numerous narrations — indeed, the Ḥanafī scholars hold it to be obligatory. Furthermore, its time is narrow and limited, expiring with the passing of the days of Eid, whereas the 'aqīqah may be delayed until one becomes financially capable of performing it.
And Allah Almighty knows best.
Can Zakah or Zakat al-Fitr be given to a poor brother?
Yes, it is allowed to give Zakah or Zakat al-Fitr to a poor brother, as he is considered among the eligible recipients.
What should a person who doubted having missed a pillar of prayer do?
All praise be to Allah, The Lord of The Worlds. The prayer of one who doubts having missed one of its pillars is valid, and nothing is due on him/her since having a doubt after offering the act of worship doesn`t undermine its validity. And Allah Knows Best.