How does one go about doing a Ph.D. in Europe as an American?

I am not even sure it makes sense to speak about “graduate schools” in Europe. In the countries I know, the big divide is between the master's degree and the PhD program, and not between bachelor and master programs. A master's degree is mostly another diploma with slightly higher requirements/more focused topic but otherwise not unlike a bachelor's degree in the way it's organized. There might be some exceptions here and there but it would be highly unusual to enroll in a PhD program without a master's degree.

Since tuition fees are relatively low in many countries, funding is not an issue in the way it is in the US (you still need to pay for the costs of living obviously but that's already the case for a bachelor's program). In some countries (e.g. France), people coming from abroad to study are treated exactly the same (which means paying something like EUR 250 plus some money for health insurance and a few other things), in others (e.g. the Netherlands), they have to pay a much higher fee (EUR 13000 per year where I work). In Germany the situation seems to be very fluid, with the rules set at the provincial level and changing all the time but I think fees are at most EUR 1000 per year.

Because local students don't have to pay that much and the fees don't differ much if at all from one program to the other, it's not surprising that you didn't find information about whether the master's program is “funded” or not, it's not a distinction that makes sense at this level. Furthermore, in many countries, support for students who face financial hardship is available from the government and not through the universities. Either you qualify and you can choose the university you want or you don't and you have to pay but you wouldn't specifically look for a “fully funded” program.

After the master's degree, the status of PhD candidates also varies a lot from country to country. In Switzerland, Scandinavia, the Netherlands and increasingly Germany, you are an employee with a reasonable salary (people will tell you it's less than what professionals with similar qualifications get, which is true, but you can still live comfortably). Some form of funding (from the main university budget, a research grant, European grant, corporate sponsorship, foreign government grant, etc.) is therefore a prerequisite. In France or Italy, you are considered a student and working conditions are often poorer. In STEM fields, PhD positions are usually funded and you do get some money and resources for your research. In the humanities, it's not uncommon for PhD candidates to have no funding, sometimes not even a desk and to scrap a little money to get by through teaching or even another job.

Here again, tuition is usually not the issue but living costs are (for at least three years, at a time where you might want to start a family, etc.) Even an unfunded humanities PhD candidate in France does not have to pay much to the university. Unlike bachelor's or master's degrees, I don't know any university where PhD candidates from abroad would have to pay more to be admitted or be categorically barred from some funding (but the situations are so diverse that it might exist somewhere I guess). The only hiring restriction of that kind I know are research fellowship from the European Space Agency.

I'll start with the bad news:

Does anyone know of any good resources so I can figure out this situation?

Such a resource does likely not exist, as the answers to your questions will vary widely between different european countries and even different universities. Europe is historically a hotchpotch of different academic systems.

I will try to give some answers that I think are true in most places. For all of the answers, you will likely find exceptions and differing systems if you look long enough:

Are Master's degrees in Europe funded?

Unfortunately very rarely.

That being said, as aeismail points out, most european degree programs are either for free or (compared to US programmes) extremely cheap. That still leaves you with costs of living and opportunity costs, though.

And what would happen if I would decided to get a Master's degree in Europe but then come back to the United States for the Ph.D.?

I don't quite understand that question. You would have a master's degree, which would allow you to either find a job or do a PhD in the US (however, you could have done both with a bachelor's degree as well).

Is it possible to go straight into a European Ph.D. without a Master's at all? (I'll be graduating with 18 credits of graduate coursework and 27 credits of independent research).

Usually this is not possible. I have heard here on academia.SE that there are universities that make exceptions, but the places that I am aware of do generally not allow to start a PhD at all without a master. However, not all is necessarily lost. Some places (including my current university in Zurich) have established compromise solutions for incoming research students without master's degree. Here, for instance, a professor can apply that you do a research-based "fast track" master, which allows you to skim on the course work and do research with the professor instead of most of the regular master curriculum. This is quite comparable to the US system, but as far as I know it is node widely publicized (I don't think this possibility even appears anywhere on the web page).

I can not speak for all of Europe but in Denmark the PhD programme is a job. You are hired by the university to conduct research so you will get paid. You need a Master's degree to apply for a PhD position, and there are probably some grade point average you would need.

Here is a link to a .pdf document with the danish legislative framework.

You can apply for the PhD programme even though you have taken your Master's degree in USA. If you want to take your Master's in Denmark you need to apply for it through your own university if you want it to pay the stay. There is also a possibility of getting paid while in Denmark by the governments state education support.