Criminally negligent homicide, and manslaughter are specific laws. Conviction generally requires proof that the subject's actions were the proximate cause (a legal term with its own case law) of the victim's death.

I'm aware of only one case where someone was convicted of such a crime in the US without being physically involved in the death: Michelle Carter, who directly and repeatedly encouraged her boyfriend to kill himself, and goaded him into continuing what was ultimately a successful suicide attempt when he started to back out. Despite her active encouragement and unambiguous intent, the legal theory was controversial and the case has seen multiple appeals.

I find it quite unlikely that a court will accept the argument that intentionally making someone sad is the proximate cause of their death by suicide, even if done to a large number of people at the same time. Were that argument accepted, it could be applied to other situations affecting the emotions of many people just as easily, such as producing a sad song or movie.

Indeed. And note that these laws are generally state laws in the US, not federal. So just in the US, there's a lot of possible versions of the law/jurisdiction Facebook could face.

I didn't think if that, the jurisdiction would be where the person died, so potentially anywhere in the world with an extradition treaty could charge them.