I know that all though the seperation is pretty de facto and well respected, many countries, such as the UK, have official connections between church and state. Do most? Which do and do not? What sort of connections are there?
France is an intensely secular country with a well defined SoCaS.
In Germany, on the other hand, citizens pay a tax which goes to the church.
Every sort. France, for example, is a secular republic with a long history of “laïcité”, but the state owns all churches built before (I think) 1904, and licences their use by various religious denominations. And it has recently prohibited the wearing of prominent and distinctive religious garb in public schools. Both of these measures would be unthinkable in the US tradition of separation of church and state. Germany and other countries collect taxes on behalf of the churches,and pass them on to the churches. They do this on behalf of several churches, but there are some religious denominations which they refuse to recognise for the purpose. Denmark and, I think, some other Scandinavian countries have a formally established state church – always Lutheran - which in many respects (including funding) is treated like a department of government. Most of the population rarely if ever goes to church, but establishment does not seem to be politically contentious. *Italy{/i} has a treaty with the Holy See which accords special privileges to the Catholic church. Ireland has a constitutional prohibition on the establishment of religion, but this does not prevent the state from funding church-sponsored schools. Uniform rules are applied, and the schools of several different denominations are supported in this way, as well as at least one Jewish school and one Islamic school. But in all of these countries religion plays a much smaller part in public political discourse than it appears to in the US.
Basically there is an astonishing diversity of approaches to this question.
In addition to the church tax, Religious education is a regular subject in German schools. Usually there are classes for Roman Catholics, the (Lutheran/reformed/united) Evangelical Church in Germany representing the vast majority of protestants and “Ethics” classes for everyone else. The teachers for the religious classes are paid by the states like all others but require approval of their church.
In the states the churches enjoy certain privileges related to the media. They have seats in the ratings agency and in some committees which control the public broadcasting. (Note: We have nudity in primetime over-the-air TV 
)
Which church? Catholic or Protestant? Or whichever has the most parishioners in that particular state? The latter would seem like a throwback to (or holdover from) the “Protestant if the Prince wills it” days of the Reformation.
Whichever church you belong to. The states collect the tax and so long as your church (synagogue, mosque, whatever) has official recognition in that state, they’ll get the money. You can always leave the church but it will mean you can’t have a religious ceremony in Germany with that church. I don’t care what happens when I’m dead, not the marrying kind and would only have a child christened if it were a big deal to my partner so I opted out.
It never struck me as all that controversial or divisive though and this might be due, in part, to the fact that the tax doesn’t amount to all that much.
Sweden had a state church, but it was dissolved in 2000. Norway still has a state church, but there’s talk about dissolving it here too. There’s a governement comittee working on the issue which will give a recommendation next year.
Government gives funds not only to the state church, but also to other societies of belief and non-belief, including the Humanist Association and the Heathen Society.
One result of having a state church is that the church here is, in some ways, more radical than it might have been without its marriage to the state. Bishops are appointed by the government, and some highly controversial appointments have resulted in both female and pro-gay bishops.
What about German atheists, or pagans? Do they have to pay anything?
Nobody has to pay anything. You can opt out of the church tax if you wish and, while this may cause friction with the church you belong to (assuming you wish to belong to a church) it’s not a matter that concerns the government one way or another.
Some countries that have a church tax system offer the alternative of paying an equivalent tax to be directed to a secular ethical or benevolent group. Some offer the alternative of not paying the church tax at all. And some, I presume, offer both.
If I’m not mistaken, Iceland has a state church and aslo operates a church tax system which funds all churches/ethical bodies on the same basis - i.e. each taxpayer can direct where his or her church/ethical tax goes. Whether the state church gets any financial support over and above that, I don’t know.
Finally, it’s worth pointing out that some churches refuse to participate in the church tax arrangements, encouraging their members to opt out of paying any church tax and instead make contributions directly to the church. Other churches which have been refused recognition (the Scentologists?) have no option but to pursue this course.
Except for those who don’t. There’s a “no thanks” option. At least there was when I paid tax in Germany.
The Danish State Church (Lutheran) is indeed funded by tax money (again, you can opt out) and is the most watered-out, luke-warm inoffensive religious organization imaginable. It is definitely a case of the State running the Church. So while I disagree in principle, it’s most certainly a way to keep the clergy in line. There is one (1) person who can’t opt out of the church: The reigning monarch. For the rest of the Danes, thety can use the church or not, just as they can support it or not.
Complete state and church separation is perhaps a little hard to do in many European nations, because centuries if not millenia of seriously important cultural history is to be found in the churches and convents. The old cathedrals, manuscripts, vestments, antique chalices etc. etc. are considered a national treasure (in a completely secular sense) as much as a religious one. And as such, they come under the Government’s protection.
I find it rather fascinating the role religion has in European countries. I only wish that perhaps the US religious establishments could avoid stepping on other people’s toes the same degree like how they do things in Europe.
For the record, we can choose if a small part of our taxes goes to a church or the University. If you formed your own church, you could even get the money yourself; this was all the rage until some wise-ass tried to form the Atheist Church, forcing the government to make stricter laws regarding this matter.