Which Orthodox Churches Accept the Baptism of Heretics?

This controversy is as old as the 250s AD. Cyprian wrote:

... since there cannot be two baptisms, if heretics truly baptize, they themselves have this baptism. And he who of his own authority grants this advantage to them yields and consents to them, that the enemy and adversary of Christ should seem to have the power of washing, and purifying, and sanctifying a man. But we say that those who come thence are not re-baptized among us, but are baptized.

At that time, the North African bishops did not believe in accepting the baptisms of heretics, but Stephen bishop of Rome told them they had to. They actually held a council to condemn him, calling him a "Judas." Yet, Stephen eventually won, and it became standard policy to accept any Trinitarian baptism.

I'll answer your two questions simultaneously. Trinitarian baptism is canonically valid, always. Constantinople, Alexandria, Antioch, and Moscow Patriarchate within Eastern Orthodox communion accepts the validity of baptism under the Trinitarian formalism. Because the validity is rested on the formula not on the professed faith.

St. Athanasius initially accepts Arian baptism, they just need to confess Nicene faith and do penance before received communion and/or restored to their canonical rank. But later as Arians started to develop their heresy further than what Arius started, Athanasius began to reject the validity of their baptism not because Trinitarian formalism is void of efficacy when performed by Arians but because they changed the Apostolic baptismal formalism.

For the Arians do not baptize into Father and Son, but into Creator and creature, and into Maker and work. And as a creature is other than the Son, so the Baptism, which is supposed to be given by them, is other than the truth, though they pretend to name the Name of the Father and the Son, because of the words of Scripture, For not he who simply says, 'O Lord,' gives Baptism; but he who with the Name has also the right faith. On this account therefore our Saviour also did not simply command to baptize, but first says, 'Teach;' then thus: 'Baptize into the Name of Father, and Son, and Holy Ghost;' that the right faith might follow upon learning, and together with faith might come the consecration of Baptism.

St. Athanasius, Against the Arians, 2:18:42.

Because of this some fundamentalists erroneously insist that baptismal efficacy rested not on Apostolic baptismal formalism but on the confessed faith. This is why Greek Orthodox Synod of Resistance and Russian Orthodox Church Abroad (not in communion with Moscow Patriarchate) are plain and simple committing the heresy of Donatism.

When Arians began to have baptized in the name of the Creator and creatures, they changed the Apostolic form of sacramental baptism. The Anomaeans, a further development of Arianism, baptized with the new formula:

In the name of the uncreated God and in the name of the created Son, and in the name of the Sanctifying Spirit, created by the created Son

St. Epiphanius of Salamis, Against Heresies, 77.

This is why the efficacy of baptism is not founded on the professed faith but on the Apostolic formalism. When a sect changed this sacramental form their baptism is invalid. Baptism in the name of Christ only is also canonically valid but I will not discuss it here unless asked.

Baptism is the only sacrament which, in the absence of a priest, may be administered by the layman (man or woman) by virtue of the universal Christian priesthood. ... All Christian baptism conferred in the name of the Holy Trinity is valid, not only within Orthodoxy, but outside its limits; the practice of the Church, ancient and modern, is sufficient testimony to this. As a general rule those who have been baptized once are not re-baptized....

Fr. Sergius Bulgakov, The Orthodox Church, 1988, p. 112.

To see whether or not New Calendarist's baptism is valid or not, we can see a simple case from the baptism of Emperor St. Constantine the Great by a semi Arian Eusebius of Nicomedia. During Nicene council out of 320 bishops, two refused to sign Nicene faith: Theonas of Marmarica and Secundus of Ptolemais. From 318 Nicene Fathers, three who signed the Nicene faith refused to excommunicate Arius: Theognis of Nicaea, Maris of Chalcedon, and Eusebius of Nicomedia. All five were deposed and exiled. But later the Church canonically exonerated all three who signed Nicene faith plus Arius who then accepted Nicene faith but died by poison because of politic before received back his canonical rank. Despite of Nicomedia's past history and his allegiance to Arius before, during, and after Nicaea, the Church unanimously accept his baptism of Constantine to be canonically valid whether or not they're New or Old Calendarists for both venerate him as a great saint of Orthodoxy.

I disagree with the many Orthodox who claim a non-Orthodox Baptism is merely an empty shell which is filled with grace only after being Chrismated into the Orthodox Church. Indeed, Orthodox call Emperor Constantine (d.337) as "Saint and Christian", but he only accepted Baptism near his death, and then it was done by a heretical Arian bishop...leading one to understand that "economia" (a phrase Eastern Christians use which means "through God's grace for the good of the whole Church") gave Constantine's Baptism validity and grace.

Daniel Joseph Barton, My Belief in the Immaculate Conception: Part 1.