Is it obligatory to slaughter one ram or two in the Hajj for each of the spouses? And is it permissible for the spouses to offer the sacrifice in their home country?

Firstly: If you are performing Tamattu' or Qiran Hajj, then each of you must offer an independent sacrifice (hady). One ram is not sufficient for both of you because the blood sacrifice for Tamattu' and Qiran is obligatory. Whoever cannot afford it must fast ten days. However, if you are performing Ifrad Hajj, no sacrifice is required from you.

Secondly: The Udhiyah (sacrificial animal offered on Eid al-Adha) is not prescribed for the pilgrim; rather, the Hady (sacrifice for Hajj) is prescribed for him. If the pilgrim is alone and his family is in his home country, he should leave them money to buy an Udhiyah. Thus, he offers the Hady while they offer the Udhiyah, because Udhiyah is prescribed in cities, whereas in Makkah, it is the Hady.