Thank you for showing interest in the various Eastern Catholic Churches!
The Melkite Church came into Communion with Rome in the 18th century as a decision by the Patriarch and the Synod of Bishops. At that time some Melkite Bishops decided to stay with Constantinople, and became the Antiochian Orthodox Church with their Patriarch appointed by Constantinople.
As for Churches like the Coptic Church, the Monophysite controversy has been resolved between the Catholic Communion and the Oriental Orthodox. Their theology was always orthodox, but there was a misunderstanding of terminology used. The Copts actually anathemized Eutyches, the author of Monophysitism, on their own shortly after the split between themselves and the Chalcedonian Churches. There was no reunion at that time, however, because they were concerned that the language of the Council of Chalcedon was Nestorian.
So Churches like the Coptic and Syriac and Armenian are now considered fully orthodox in their Faith, and there is no restriction placed on them entering the Catholic Communion. In fact, we even have limited inter-Communion with the Syriac, Armenian, and Indian non-Chalcedonian Churches (there are standing agreements between these Churches and the Catholic Communion that their Faithful and ours can received Sacraments from the other Church in certain circumstances such as weddings, or when there are no priests of their own in the area).
Peace and God bless!