Good or bad Musk wants to open the platform up. Whether that will just change the speech to what he likes or actually broaden it we'll see.
“the more an owner, for his advantage, opens up his property for use by the public in general, the more do his rights become circumscribed by the statutory and constitutional rights of those who use it.” (Marsh v. Alabama, 326 U.S. 501, 506 (1946)
They have since narrowed their interpretation somewhat, but some states (including California) have ruled that de-facto public gathering places such as shopping malls are areas where free speech is protected, even though they are privately owned.
So generally, yes a private platform can do what they want, but if their primary focus is to be open to the public, they may be more bound to providing first amendment protections.
AFAIK no court has specifically ruled on whether online public spaces should be treated the same as physical public spaces.
A culture of free speech is just as important as a law prohibiting Congress from abridging it. It is true that private companies have their own free speech and private property interests they should safeguard, but that does not mean they cannot do more to raise a culture of free speech on their platforms, and they should!
Government, no question about it. Twitter doesn't throw you in jail.
It is not. The first amendment of the constitution of the United States is, but freedom of speech is just freedom of speech.