What is the ruling on someone who possesses gold with the intention of saving/hoarding it, and owns nothing else — is he obligated to sell a portion of it in order to pay the zakāh due on it?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
Zakāh becomes immediately obligatory on cash, and likewise on gold — that which is not designated for a woman's personal adornment— once both reach the niṣāb (minimum threshold liable for zakah) and a full ḥawl (lunar year) has passed over them. It is not permissible to delay payment when it is genuinely possible to pay — meaning, when the wealth and the eligible recipients are both available. It is stated in Mughnī al-Muḥtāj (Vol.2/p.129): "Zakāh is obligatory immediately once it becomes feasible to pay it, meaning when the wealth and the eligible recipients are both present."
The one paying zakāh must discharge the obligatory zakāh due on saved (hoarded) gold either from the gold itself, or from its equivalent monetary value, in the event that no other funds are available — since paying from the gold itself is the original default (al-aṣl). It becomes permissible to pay from other funds instead only because zakāh is built upon the principle of ease and leniency, as stated in Asnā al-Maṭālib (Vol.1/p.366): "When a full ḥawl has elapsed on wealth other than trade goods... zakāh attaches to the actual asset itself (al-‘ayn), and the poor effectively become partners in it — even, for example, [becoming partners] in camels through the value of a sheep [owed as zakāh on them]... Paying from other wealth is permitted only because zakāh is built upon the principle of ease and leniency."
Accordingly: the one paying zakāh must discharge it either from the actual saved gold itself, or by selling a portion of the gold equal to the amount of zakāh owed. He may also borrow money to pay the zakāh, or pay the zakāh in installments before the ḥawl is complete [as an advance payment]. And Allah, the Exalted, knows best.
Is the son of the sister by breastfeeding considered a mahram in Islam, and is it permissible for a woman to go for Hajj with him?
Everything that is prohibited by lineage is also prohibited by breastfeeding. The son of the sister by breastfeeding is considered a mahram, just like the son of the sister by blood, and it is permissible for him to be her mahram for Hajj and Umrah. And Allah Knows Best.
Is it permissible for the wife to give the Zakah (obligatory charity) of her money to her poor husband?
Yes, it is permissible for the wife to give from the Zakah of her money to her poor husband. And Allah Knows Best.
Does the intention (niyyah) of the principal (muwakkil) suffice for the sacrifice (dhabḥ) of the udhiyyah?
Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
If a person appoints an agent (wakīl) to perform the slaughter (dhabḥ) on their behalf, the intention (niyyah) of the principal (the one appointing the agent) suffices, and there is no need for the agent to have this intention himself. In fact, even if the agent is unaware that he is performing the sacrifice on someone's behalf as an udhiyyah (sacrificial offering), this does not affect its validity. And Allah, the Most High, knows best.