I do exactly what it is you are talking about doing: Web application development and design for customers spread around the world. I have a visa to live in Germany and to work freelance for German companies but not as a fulltime employee. I approached it with very similar rationale as yourself (bringing money into the city can't be a bad thing, etc...) To get the visa I provided a well-written business plan (ca. 10 pages, translated to German) describing how I was going to make it here, proof of sufficient cash on hand to cover all my costs for well over a year, demonstrable German language skills, and two letters of intent to hire from Berlin-based companies. 200 euro and 7 days later with the help of an immigration lawyer I had the visa I wanted. Painless and cheap, when compared to some other countries. The visa requires renewal through 5 years at which point one can apply for permanent residency. Thankfully my German was good enough that I didn't have to take any integration courses.
I hear a variety of stories from people so I don't know what the norm is, if there is such a thing, but it's definitely possible to get what you're after.
I have private health insurance with dental, full coverage no deductible, from ALC-Allianz. I believe John G is "da man" in that category. Though you can also get it through Mike Woodiwiss at Spectrum (use the search) It costs me around 130 euro per month, which is peanuts compared to US insurance costs. I actually just went to a doctor last week first time so I'll soon find out how good the service is.. Some salespeople will try to tell you ALC is not sufficient but they are wrong."