It's the "hate speech" label that gets complicated based on who defines it. Many who claim things are hate speech today also use hate speech themselves (following your definition of suppression based on race) to shout down and ban others.
It shows just how complex this situation is and why "hate speech" is precisely the type of speech that needs defense in the first place because it's mostly rooted in defiance and disagreement, not outright hate.
No. The 1st amendment of the US constitution is about government interference - the idea of free speech is broader than that.
Yes, it does. It has to, otherwise free speech is defined as any arbitrary level of censorship a government allows, and one could say North Korea has just as much "free speech" as the US.
Now, I'm not making an argument that free speech as such is preferable or even feasible in a civil society, or that censorship per se is wrong, and I'm definitely not making the typical slippery slope argument that any form of censorship will inevitably lead to tyranny.
However, free speech has to be defined as a constant to mean anything objective.