"hate speech" is protected speech, but you have to be careful who your judge is. There is the law as it is defined by the scotus, then there are judges who politic from the bench and judges aren't well educated who will scream probably cause exists any chance they get. There are generally 3 types of of analyis of free speech. One is the traditional categories of speech, another is a captive audience, and a third is a time manner or place restriction.
For the most part what private companies allow on their own platforms is not covered by the first amendment. Imagine opening up a server for the christian community, and some troll comes on and keeps posting trans por*. There are strong arguments to support the property rights of the owners. Although, increasingly, that could change. California, where many of these dotcoms are, would be an interesting place to start since they extend the first amendment into private malls subject to time manner and place restriction. In addition, when a public official blocks a user on social media that can open the public official up to section 1983 suits. If twitter were to intrefere in that access to contacting a public official, that a person could try to plead that twitter is violating their first amendment rights. Although I think such a suit is untenable.
Untenable because Twitter not only would say they are a private company, that as a private company, unlike the public official, they can't be held liable for speaker/content based discrimination as the public official can, and they could argue that there exist alternative remedies such as an office trip, the us mail, a email to the congressman's webpage and so on.
The alternative channels are time manner and place restrictions, and there must be some compelling governmental interest for their policies. Such as preventing noises when most people are sleeping, for city beatification, safety regarding road construction, sidewalk traffic, safety/environment at large events,and so on. The policies must be content neutral and they must leave an adequate channel of communication. When their is a less speech restrictive means the government must choose that alternative. It doesn't mean that the government has to choose the best alternative.
The internet is not a captive audience. See janet reno v united states, also see bolger v young drug products co. People can tend to have to actively look for offensive content, and supposive they do see it they can avert their eyes. They can block social media users, they can throw away emails, they can block webpages and ads all without the need for government intervention. The home, like medical centers, the courts, the schools, and your travels to and from work (rarely used) are examples of captive audiences. The home being a captive audience are why the 7 dirty words can't be spoken on the radio during day time, because someone could manually turn a dial (how old the fcc v pacifica foundation case is) and hear unwanted words in his own home against his wishes. It is also a similar basis, see Rowan v us post office department, why a home owner can block certain commercial mass mailings that he find arousing-even a dry goods catalog. However he isn't able to do so, short of abusing the family courts, with ordinary mail-including commercial informative mass mailings (central hudson, I think it was).
The third method would be the categorical approach. That would include true threats and their subcategory intimidation (virginia v black), fraud, libel, fighting words spoken face to face (gooding v wilson), obscenity/cp (miller v california/new york v ferber), conduct integral to criminal conduct (giboney v empiror storage), and finally incitement (brandenburg v ohio)
Freedoms of speech and press do not permit a State to forbid advocacy of the use of force or of law violation except where such advocacy is directed to inciting or producing imminent lawless action and is likely to incite or produce such action.
-brandenburg v ohio
As far as suing twitter, there may be a theory that I haven't researched. It wouldn't really be a lawsuit against twitter per se, but a twitter and various john does. If liberal actors and organizations (cough CNN) are pressuring twitter not to do business with conservative speakers, that could invoke liability under a secondary boycott.
I have been trying to restart an old social network I wrote. I need to get back to that, but I got aggravated after the zindex refused to work properly (some css attributes messes up the hierarchy I hear) and took a break and forgot to come back. Originally it was html based and supported user entered javascript, but 10 years later I see that as a security risk. So a visual interface and script that compiles into html and javascript and hopefully bypasses the security risks.
RE: Should We Ban Incitement to Violence?