Is it obligatory for the questioner to slaughter the Udhiyah, given that he is newly married and his father has passed away, while his elder brother and his sister's husband perform the slaughter?

The Udhiyah (sacrificial offering) is a confirmed Sunnah, not obligatory, according to the majority of scholars. You are not required to offer a sacrifice even if your brother does not. If you live with your brother and he provides for you, then one sacrifice is sufficient for both of you. Otherwise, each of you should have a separate sacrifice.