Explicitly banned by the Catholic Church, in fact. Though the trade continued illicitly in a small way throughout the period; and non-Christians held slaves, so that many Christians found loopholes through which to acquire slaves in Italy and Spain. However, slavery was quite uncommon in Catholic Europe; I honestly have no idea what the Byzantine position on slavery was.
Not true. While the Church gradually and over time succeeded in largely banning the enslavement of Christians by other Christians, enslavement of non-Christians remained permissible throughout the middle ages and beyond. The Papal States used to use Muslim slaves in their galleys. Even the ban on enslaving Christians was not consistently applied throughout the period.
