No, "everyone" does not need to simmer down.
No, the right does not "do it too." Not as extensively and not with the same viciousness.
No, Sarah Palin's crosshairs map was not an invitation to shoot Gabby Giffords. Everyone who is not an imbecile understands that it was a call to vote Giffords out of her office using a metaphor that was and is thoroughly typical of American political parlance. And while we're at it, cease the autistic screeching about the occasional Obama effigy. Though that is indeed an example of violent imagery targeted at a sitting president, such effigies were never hoisted by conservatives of note and were never condoned by the same.
The right was intemperate at times in its opposition to Obama. There was a lot of nonsense floating around that suggested he was not born in the U.S. and was secretly a Muslim. (Personally, I think he's just a practical atheist wearing a Christian skin suit, but that's neither here nor there.) There was talk of his being a tyrant -- and a traitor to boot. (Whereas I think he's just a hideously misguided red diaper baby who, yes, abused his power on several occasions through a lack of understanding of our Constitution.) And yes, in certain dark corners of the internet, there were trolls wishing for his assassination.
But overall, the Tea Party Movement and its offshoots, for all their flaws, were peaceful. I don't recall any incident in which a Tea Party protest devolved into a destructive riot. On the contrary, when Tea Partiers demonstrated against Obamacare and excessive taxation in general, they left their rally grounds cleaner than they found them. The vast, vast majority of these folks didn't seek out Obama supporters to harass or cudgel for disagreeing with the Tea Party's aims, and they didn't make excuses for anyone who did misbehave.
The same cannot be said about the left. While there are many on that side of the aisle (like the center-left liberals I follow on YouTube) who absolutely abhor political violence, there are also many who don't -- and the members of that faction are not as marginalized as their analogs on the right. In fact, the thuggery of Antifa and BLM protestors has been repeatedly rationalized by leftwing writers with sizable audiences.
The rightwing blogosphere has never seriously entertained the idea that it's okay to punch commies who've done nothing but espouse their lunatic views.
I have never seen a rightwing group flood a college lecture hall to shout down a speaker the right doesn't like. I've certainly never seen a rightwing group set a campus on fire to prevent a leftwing speech or teach-in from going forward.
I've never seen a rightwing celebrity pose in a photo shoot with a mock-up of a leftwing politician's severed head.
I've never seen conservatives endorse the harassment of ordinary leftwing Americans who are just trying to go about their business. No rightwinger has ever videoed himself screaming and hollering at a total stranger for displaying a Clinton campaign sign on a front lawn - or selling an LGBT flag in a discount shop - because he was confident his fellow rightwingers would think his tantrum was "awesome."
Rightwingers don't hit leftwingers first. They respond.
There is a difference between the left and the right -- a very real difference. The two sides are not equally at fault for our (currently) cold civil war, and to claim otherwise is utter foolishness.
Update: It's just come over the wire now that two Trump supporters have rushed the stage at the Shakespeare in the Park rendition of Julius Caesar in New York City. So, okay -- rightwingers now have one attempt to use the heckler's veto on their record. I think my general argument still stands, however. We're still looking at a difference in degree large enough to be considered a difference in kind.
Further Update: Please don't make a liar out of me, conservatives. I understand that you are frustrated by the asymmetry I described in the original post above. I understand that you're sick of being held to high standards while the left is getting away with bloody murder. I am too! But it's still wrong to try to silence the other side to "give them a taste of their own medicine." Surely we can think of a more creative way to inflict pain for bad behavior that does not involve jettisoning our principles. Please, for the love of God and free speech, use your imagination.
Update III: Yeah, what Nicki said.