Does the word 'ecclesiastical' specifically refer to the Christian church or is it commonly used in the context of other religions?
It refers to the Christian Church only. See the Oxford Dictionary definition. Try clerical for a more general synonym that can work for all religions.
Yes. Only Christians.
It comes from the Greek Ekklesia, which began to be used by the earliest Christians to describe their collective identity in terms of their rituals and customs and structures.
But earlier Greek speaking Jewish people used the term to speak of Isreal.
The etymology of the term comes from the Greek meaning the gathering of an assembly to make political decisions.
But Christians made way more use of it, making it more constrained while at the same time, more complex. In fact, christians have preserved some of the original meaning — an assembly of Greeks to make political decisions — and took it to make moral and religious decisions for the masses; hence we have Christian evangelicals in the forms of mission groups and and not to mention the ancient crusades.
Christianity: the First 3000 Years by Diarmaid McCulloch.