Is it obligatory for each of the two brothers, who live in the same house and share food and drink, to offer a sacrifice (Udhiyah), or is one sacrifice sufficient for both of them? And is it permissible for the other [brother] to shave his head and trim his nails if his brother offered the sacrifice?

As long as you both share in the expenses, then, according to the Malikis, each of you must offer a sacrifice (udhiyah (definition)). However, the Shafi'is and Hanafis are of the opinion that a single sacrifice is sufficient in your situation. It is sunnah (definition) for one who intends to sacrifice not to cut their hair or nails until after the sacrifice has been slaughtered.