Your description of the issue is really misleading, Rare. The mosque was not holding a Muhammad cartoon contest; it was a freaking mosque, they're against that kind of thing. I'm not sure how you got that idea.
The actual events behind the protest and hashtag were that on May 3rd two Muslim men went to a Dallas suburb and opened fire on a building where the "Muhammad Art Exhibit and Cartoon Contest" was being held. They shot one unarmed security officer in the leg before police who were helping with security returned fire and killed them both. This armed protest was held at a mosque in conjunction with another Muhammad drawing contest staged
outside the mosque, and it was done in response to that attack in Texas. This particular mosque was chosen because it was one that the shooters had used before they decided to go shoot up the event in Dallas. Context is important.
That out of the way, I think the armed part of the protest was a bad idea, just asking for trouble, and I'm surprised nobody ended up getting hurt. Protesting at the mosque was also in poor form, as it would be the same as going and protesting at some church after a couple nutjobs who happened to go there went and killed people in the name of god, which is in fact a thing that still happens in this day and age because crazy people of all faiths often say that god/scripture told them to do horrendous things. It would have been more constructive to do an unarmed protest elsewhere, perhaps around the event center where the shooting took place, rather than frightening and inconveniencing a bunch of Muslims who just happened to attend the same mosque as the shooters.
The plain fact that people were holding a protest and drawing Muhammad in response to the attack though? I fully support their right to do it. It has nothing to do with racism because, fun fact, Islam is not a race. Just as Christianity is not exclusive to one race, Muslims are a diverse lot rather than a singular race. It is a religion, an ideology, and all ideologies are open to criticism and hatred. Muslims may not like that, but tough shit for them, if they want to live in the United States then they are subject to the laws and freedoms that entails. One of those freedoms is the freedom of speech, which means that the government cannot stop someone from expressing their thoughts and beliefs unless they are inciting violence or other criminal acts. Art is a protected form of free speech, therefore cartoons of Muhammad are 100% fine according to United States law. Are the people drawing those cartoons a bunch of dicks for purposely going out of their way to make Muslims upset? Absolutely. Is it hate speech? A lot of it probably was, but luckily the United States is one of the few countries left that does not have foolish laws against hateful but not harmful speech. However, despite what I think of their opinions, as the quote often
misattributed to Voltaire goes, "I disapprove of what you say, but I will defend to the death your right to say it." Freedom of speech is one of the core values of the US and most of the western world, and I cannot support those who would try to cripple it just so one special class of people can prosecute those who offend their sensibilities. The protestors did nothing wrong, they're jerks but being a jerk is not and should not be a crime, end of story.
To be honest, I find some of those on the side of the hashtag, those counter-protesting the anti-Islam protest, to be the more troublesome lot at the moment. Most of them just held opposing views and expressed that, no problem there, that's just as protected under freedom of speech as the anti-Islam messages are. However, upon skimming through the hashtag on Twitter I found a lot of calls for the anti-Islam folks to be silenced or punished for what they had done. Those people are the real problem, as far as I'm concerned. Trying to fight hatred and prejudice is cool, but wanting to do so by way of undermining civil liberties is really stupid. I could rant for a long while about how much I dislike the authoritarian wing of the progressive left that pushes for this kind of crap, but I'll just leave it at that. Also, before anyone tries to label me as a conservative for that statement, nah, I'm pretty liberal and progressive but also rather anti-authoritarian.
TL;DR the anti-Islam protestors are dicks but being a dick isn't illegal or really harmful, I've got no major problem with them.