Living in Germany, working remotely for a Swiss Company- taxes?

I am an EU citizen, living in Germany.
I was offer the possibility to work 100% remotely for a company in Switzerland, a "classic" employment contract. I will not be an Grenzgänger. I understood that I need to pay taxes were I work, so in Germany.

My questions are:

1. Do I need to pay remote work tax contributions or any other tax in Swiss system?

2. Who is paying the income taxes and the social security in Germany, me or the Swiss employer?

3. I have a health insurance in Germany. I assume that I don’t need a Swiss health insurance due to remote work. Am I right?

Thank you for your time.