How health insurance system works in Germany?

You asked about many different aspects of German health insurance.
In my reply, I will not talk about who can or cannot join which of the two systems (public or private), as this is very complicated and was covered in previous posts on this forum.
Regarding payments, a publicly insured person just shows his/her insurance card when visiting a doctor or clinic, and the costs are directly charged to the insurer. A co-payment by the insured person is only required for some non-standard treatment (e.g. certain dental items), for hospitalization (I think it's EUR10/day) and medicine (EUR5/item).
A privately insured person gets the bill directly from the doctor, has to pay it him/herself and apply for reimbursement from the insurer. Depending on the tariff (there are hundreds at any level of coverage) you'd get all or only part reimbursed.
EUR350/month is cheap for a private insurer and most likely covers only yourself. (Unlike in the public scheme, family members without own income CANNOT be added at no charge at private insurers.)
The employer pays roughly half of what a public insurer would cost, regardless of whether you are privately insured or public.

Ok. Thanks for the reply. Its informative.