German health insurance is compulsory for anybody on a residence visa. That includes student visa. Students currently pay the subsidized rate of approx. €80/month for the insurance (if you join the public scheme - the private scheme might be cheaper if you are young, healthy and have no family).
Edited to add: Yes, they pay for all medically necessary medication and aid, incl. doctor visits, laboratory tests, etc. There might be a small co-payment for medication, not more than a few Euro.