Should hate speech be protected by freedom of speech laws?

The definition of hate speech needs to be agreed and widely disseminated in the population before it can be penalised.
Holding religious beliefs that others see as intolerant or holding so called gender critical beliefs should not be in themselves hate speech.

no. freedom of speech does not cover speech intended to cause harm but the explicit expression of opinion.

You should be free to express your view in a clear and concise way, and for people to challenge that view

People should not be free to openly incite racist violence or make any other kind of discriminatory remark, but hate speech that is perceived to be spoken for the good of the people should be protected.

yes, but freedom of expression does not mean freedom from consequence

The ECHR provides a framework to follow then it becomes the obligation of the state to show its response is proportionate, reasonable and falls with tbe margin of appreciation, which is accessed by a court. With rights, there comes responsibilities.

No, except where it is regulated by schools, employers, or contributes to harassment or violence.