Does ill-gotten money hinder the answering of the Du`a (supplication) and the means of subsistence?
Sins hinder the answering of the Du`a, and deprive the sinner from means of subsistence; whereas, acts of obedience facilitate making a living. Allah, The Almighty, Says (What means) : "And for those who fear God, He (ever) prepares a way out. And He provides for him from (sources) he never could imagine." [At-Talaq/2-3].
Should missed Sunnah prayer be made up?
Making up missed voluntary acts of worship is from Sunnah.
Is it permissible to offer the Eid sacrifice in a non-Muslim country?
Eid sacrifice is permissible anywhere, and Islamic Sharia hasn`t determined that it should be in an Islamic country, but poor Muslims should be given a share in order for it to be valid since it is an act of worship. A portion of that Eid sacrifice should be given at least to one poor Muslim, and if that wasn`t possible then the one offering it should consume it along with his family.
Is vomit among the nullifiers of Wudu (ablution)?
Vomit does not nullify Wudu, but it is a Najaasah (impurity) that requires rinsing the mouth and washing whatever became dirty by it since the prayer becomes valid only after the removal of Najaasah. And Allah Knows Best.