With the exception of a handful of private universities most universities in Germany are either tuition-free or cheap. Whether a university will charge a tuition fee depends upon the Land (= federal state) the university is in, if they charge tuition it will be around 500 EUR per semester plus around 60 EUR membership fee in the student union.
You can find out more information about studying in Germany at www.daad.de
Note that the application process does not apply to you, as exchange student (which you would be) you do not have to pass entrance exams. Only students seeking a degree from a German university have to go through this.
There is only one problem if you only want to learn German and not enroll in any other course: You would probably be better off in a normal language school. More expensive, but easier to arrange.
You can get a residency permit for learning German at a language school just as well as for studying German at a German university, the difference will be that you won't be given a part-time work permit. Many people do just that in preparation for the DSH (=roughly: university-level German language ability) exams.
Finding a course in Beginner's German held at a German university for students actually enrolled at that university will be difficult. There are courses like that at every bigger university, but almost all of them are aimed at foreign students seeking to enroll for a full degree after passing the DSH exam.
Tip: When looking at universities note that attendance at FH's (Universities of Applied Science) is mandatory while it is non-mandatory at the "normal" universities.
I also wanted to ask about Temporary Residence Visas.
In Germany there is no big difference between the residency permits. For US-citizens with money the practise for getting a residency permit is pretty lax. Your student visa will be a "temporary residency visa" as you call it, if you are no longer enrolled in university it can easily be changed into a "temporary residency permit for language-school attendant" or such. US-citizens are even privileged, you can come to Germany with no previous paperwork and go through the application process for the student visa while already in Germany. All they want is to make sure you will not be a burden on German society, if that is fullfilled because you got the cash to support yourself without working you generally get your residency permit.