Fallacy of Division: Cops and Looters

SJW is a loaded term that's generally used to stereotype people advocating for various sociopolitical issues as militant and irrational. It's not really the name of a particular movement or anything, so it's better to just go by individual arguments without generalizing. Those look like paraphrases of paraphrases of arguments that aren't quite the same as they were originally. The original arguments aren't necessarily what you're making them out to be.

If you are against looting then you are for murderous police.

This is in the context of the discussions attempting to discredit the protests due to the looting. The statement as presented is fallacious but it could be an out of context distortion of the argument that discrediting the protests because of the looting is validating the police's suppression of said protests.

Peaceful protesting doesn't work because MLK was peaceful and they killed him.

This one is missing the context that explains how protests working is defined. It seems to be derived from responses to people citing MLK's peaceful protests as an example of how to protest "correctly" to avoid getting assaulted by police, in which case protests working is defined as not getting assaulted by the police for protesting despite doing so being a human rights violation.

Looting is a way to fight against racism

I'm pretty sure that's meant to be a claim that looting is a valid tactic in protests rather than to fight racism specifically. It's a claim without its supporting arguments, which is what would need to be addressed.

Looting is good because killing an innocent man is not as bad as looting.

That one is fallacious due to the morality of looting being based on murder being worse. It would also be fallacious to claim that anyone who says that killing an innocent man is worse is claiming that looting is good, as that's not what they actually said. The actual morality of looting is debatable.

The whole system of policing in the USA is institutionally racist, therefore all individual cops are racist.

The idea that all cops are racist is generally argued in the context that the ones who aren't are either fired by their peers for doing the right thing or resign because they don't condone those actions. Those people are arguing that remaining within a racist institution means being complicit in the racism perpetuated by that institution, hence "all cops are racist". A racist institution needs to be run by people who enforce it and obstruct attempts to address that institutional racism. The core argument to debate here is whether or not being part of a racist institution is inherently racist and why.

Overall those arguments seem to be missing context that alters what's being claimed from what it was originally. While assessing the fallacies in the versions presented is worthwhile, I felt it was important to examine the possible distortions of those points and what they may have originally been.

If you are against looting then you are for murderous police.

A classic False Dilemma. This is a common emotional argument which attempts to bounce/prompt a person into agreeing with the speaker on pain of implicitly agreeing with something undesirable, thus getting them to bypass reason. One can be against looting, while supporting peaceful protest and reforms to policing, for example.

Peaceful protesting doesn't work because MLK was peaceful and they killed him.

Red Herring. Just because MLK was assassinated does not mean peaceful protest doesn't work; his death is a distracting piece of information. Firstly, MLK is not the only proponent of peaceful protest (Gandhi, who led India to independence in 1947, also supported it) and secondly, this argument overlooks some of his achievements (e.g. Civil Rights Act of 1964) and wider influence in society.

Looting is a way to fight against racism

This is more of an opinion than a fallacy, though it is close to Failure to Elucidate since 'fight against racism' is not elaborated upon - what type of racism is being fought by looting, and how? In addition, you could argue that it is a weak form of anti-racism, since it terrifies people (especially business owners who may have been looted), which may engender hostility to the looters and in turn their movement.

Looting is good because killing an innocent man is not as bad as looting.  

Relative Privation. Like the Red Herring, it's another fallacy of relevance because the moral status of looting does not depend on the moral status of killing an innocent man. It is possible for both to be counterproductive; the fact that they are not equally so does not make looting morally acceptable or even beneficial.

"The whole system of policing in the USA is institutionally racist, therefore all individual cops are racist."

Yes, this is a typical Fallacy of Division. Since the group X has negative trait Y, all members of the group have negative trait Y. This does not follow logically. Furthermore, if we examine the use of the term "racist" here, we see a clear example of Equivocation because we move from institutional racism (racial discrimination or racist attitudes perpetrated by an organisation, such as the police, as found in policing procedures for instance) to interpersonal racism (racial discrimination or racist attitudes perpetrated by people without the use of institutional power).

It would be the fallacy of composition to say that if one cop is racist then the whole system is racist.  

Also correct, as long as the assumption that it is just one bad cop holds true.