It's before SCOTUS because a previous decision was appealled and overturned, as it should be. This would be a "limit", it would undo not only the "speech" part of the First Amendment, but the "peacable assemble" part as well. Unless you have any examples of it not being "peacable".
Bad taste? Sure, completely agree. Horrible signs? Uh huh, definitely. Protected by the First Amendment? Yep, and that's what SCOTUS will find as well.
The appeals process is not simply a matter of "I don't like mom's decision so I'll ask dad." There must be an avenue of appeal.
You're far more sure of the outcome than the SCOTUS is, as they agreed to hear the case and did not exclude it yesterday like they did with myriad other cases, which would have upheld the lower court's decision. They see merit in the avenue of appeal that has been taken.
Your understanding of the limits of the various liberties in the First Amendment is submarining your argument. Courts have also upheld that "the right of the people peaceably to assemble" does not grant free reign for any group to do anything, anywhere, anytime they wish. If it did, cities could not require permits to assemble on public land. There are limits which frame these liberties.
In this case,
Snyder v. Phelps, et al, the issue at hand is whether the precedents and arguments set forth in
New York Times v. Sullivan and
Hustler Magazine v. Falwell stand in this case. Specifically, the Hustler case protects injurious speech not based on provable facts about public figures, because even false speech "honestly believed" contributes to the greater public debate which surrounds admittedly pubic figures.
Snyder is arguing that the circuit court was wrong to extend the protection of injurious speech not based on provable facts, even that spoken "out of hatred" to a case involving a private figure instead of a public figure. At the time of the funeral, Snyder himself was certainly not a public figure. Central to the argument is the idea that while speech about a public figure may be distressing and injurious to a reputation, public figures enjoy the opportunity for public redress and recourse against their defamers while private parties do not.
Further, Snyder has argued that the attendees at a funeral constitute a captive audience. When a captive audience is involved, the government may limit the exercise of otherwise protected (free) speech, because the audience has no reasonable recourse to escape such speech. This doctrine has been upheld in many cases, and is used to prevent a wide range of distasteful and injurious speech. Snyder's argument is actually interesting and well-thought-out, as he contends that an individual at the funeral cannot avoid the hateful speech without becoming a non-attendee, just as a student cannot avoid such speech in school without becoming a non-student (hence your public school can limit speech as well).
It's not so cut-and-dried. Saying "that little thing called the Constitution" hardly frames the issues at hand.