Short answer: no. But one should define terms, especially with legal implications.

“Hate Speech” always sounded a bit Orwellian to me. Just like “Homeland Security”.

In general, I believe the jurisprudence of free speech in our country (USA) essentially says beyond, libel, slander, inciting violence, or sedition, the government can’t imprison you for expression or forcibly silence you in a public forum.

Private organizations and companies can regulate speech within their domains and property to the extent that they don’t violate other laws or rights of other parties within and without their said domains and property.

I think that’s pretty fair.

Yes. There is no contradiction. Freedom or speech is the freedom to discuss or criticise as part of a discussion, in particular the freedom to criticize those in power without the fear of repercussion. Discuss sensitive topics to all your hearts desire. Hate speech does not intend to discuss anything. Hate speech is there to target, to threaten, to belittle, to dehumanise, to attack. Hate speech is violence.

Edit; As usual with this topic “free speech absolutists” emerge, often accompanied by elaborate use of language and a thesaurus. And as usual they are not really into the entire “free speech” as in “freedom of discussion”, but rather “freedom of consequences” for themselves. Well boo hoo, ain’t that a pearl clutching shame of a slippery slope to the strawman of “who are the real Nazis” when not supporting your freedom of unadulterated hatred to run free into the world.