I think you may have trouble getting a work permit without some specific skill or profession that is in demand. Please someone correct me if I am wrong but I believe that when a company sponsors someone for a work permit they must state that they could not find a German or other EU citizen to fill the role. This would be a difficult to prove in a restaurant or cafe.
If you are under a certain age (not sure how old) I think you can get a working holiday permit for 6 months or so.
The US isn't a participant in the working holiday visa program, so that is not an option for the OP here in Germany.
Yup - loads. Everyone relishes the very real risk of a €10,000 fine for employing illegals.
Go home? That's what the authorities would prefer you did, unless you extend your visit on a student, tourist or even aupair visa. You generally need to apply for these from outside the country in the first place...
I think US citizens and citizens of some other countries can apply for these visas while in Germany.
We don't know the OP's personal situation and why she wants to do this, but I think in most cases living marginally in Germany doesn't make much sense for a US citizen. To the OP- don't work here illegally- it's too easy to get caught, you are unlikely to make much money, and there are a whole host of attendant problems, such as getting health insurance that will crop up for you. They will deport you if caught, which can happen a number of ways. BTW, the US and Germany share tax data with each other, so you can imagine what minefields that can present for someone here illegally...