It seems to me that it is part of the intrinsic nature of monarchy to be aligned with the 'national' religion. Given that the history of the English people includes the Magna Carta, the Royal Family (Prince Harry excepted) is ever committed to the English Church (read Anglican - ecclesia Anglicanum}. As such I regard it as far more likely that England will become a Republic than that the Royal Family would abandon the Anglican Church. In the affairs of humankind, I am not sure that 'forever' is as long as it used to be.
If the Anglican and Roman Catholic Denominations amalgamated as the Presbyterian and Congregational ones did, then the English monarch could not continue to be head of the Anglican Denomination. Perhaps if Francis really turns out to be the last ever Pope, the English monarch would become head of the amalgamated denominations of a United Anglo-Roman-Catholic Church. Whether that might happen and if it did, how long it might take or last, is anyone's guess. .
Legally for now it seems that they must. Charles III seems to be devout in his own understanding of things, though I know very little of the Prince of Wales faith.
Yes, but that won't stop the English from disestablishing the Church of England if they ever decide to do so in the future. Plenty of other European countries have disestablished their state church and retained their monarchy, whose royal families still worship at the former state church.