Further to what Philipp says, you can review a list of nation states in reference to separation of church and state on Wikipedia.
However, in your question you mention "freedom of religion and equality before law" and ask how other democracies have "dealt" with the problem? At what point do you consider it dealt with, or successful?
If we were being precise, wouldn't you agree that in practice, modern democracies do not successfully deliver, certainly, the "equality before the law" part?
One can simply observe current affairs in "democratic" nation states and/or take a relatively shallow look through the modern history of various "democratic" nation states for empirical evidence to support the argument that there is manifestly not equality before the law.
I would also argue that they fail on the "freedom of religion" part too. You may think this pedantic, but democracy, being a form of statism, is itself a religion, where the concept known as government is the subject of belief. In other words, the "freedom of religion" is not precisely freedom, rather it is a greater degree of choice.
One might argue that modern democracies have been more successful in attempting it, and provide greater degrees of those freedoms and equalities than theocracies, but can never actually achieve them, because whether it's a sky-ghost or a democracy, if different moral or legal rules apply to different groups, and any of them is obliged to obey any other, we have a contradiction there.