In Germany, is sexism illegal?
I cannot speak for Germany specifically but based on the jurisdictions I know sexism (or racism, ageism etc.) is not illegal.
Discrimination based on sex is illegal. Incitement to violence (aka hate speech) is illegal. Having and expressing sexist views is not illegal; however, it is reprehensible and, in relation to measurable differences in the physiology or performance of male and female brains, scientifically inaccurate.
No, sexism is not illegal in Germany.
Discrimination based on sexism is illegal in, for example, employment, business, or housing.
Sexist speech itself is not illegal, but the usual restrictions apply - insulting people is not protected speech in Germany, see StGB §185-187 (German text).
Furthermore, employers have a duty to protect their employees from discrimination and bullying by their peers, so not doing anything when an employee reports sexism can also be fined.
Most of this is covered under the General Act on Equal Treatment, which covers discrimination based on
the grounds of race or ethnic origin, gender, religion or belief, disability, age or sexual orientation