We have plenty of sufficient information to make that determination for those instances presented. What other logical explanation could there be for allowing advocating murder of white people, gang rape of Sarah Palin etc, while banning people for saying relatively innocuous things? I'd love to hear a logical explanation for those things.
In the case of Azalea Banks with her highly offensive remarks about Palin, AFAIK (correct me if I'm wrong) Banks had no prior history of this. You've been on internet discussion forums for a long while. I assume you know as well as I do that you first get warned, then temp banned, then perma-banned. Milo had a long history of being warned and temp banned before he got perma-banned. So far as "advocating the murder of white people" I cannot find that anywhere through google. Perhaps you can link it.
In any event, no, two examples prove exactly nothing. Explanations like what I've provided above for Banks hold up perfectly well when we're talking about a handful of cases. But if you have a large sample which shows apparent bias these kinds of explanations obviously would not hold. That's why you can't prove a case with two or three examples.
Explanations for disparate treatment between this individual or that in online forums can be as simple as moderator A is making the decisions in one case and moderator B is making the decisions in another. Or perhaps it is the same moderator, but one day he's in a banning mood and in another he isn't. Maybe his dog peed in his cornflakes the day he banned someone but on the day he didn't ban someone, he got an unexpected tax refund. Or perhaps one incident is reported to moderators but an arguably worse one is not. Banning patterns in online forums ARE arbitrary to some degree. I guarantee that you can find at least a pair of examples which shows an inconsistent approach for literally every online forum. But to show that this arbitrariness has a pattern of favoring one political ideology over another you need a much larger sample size.
"Research" can be used to come to any conclusion you want it to. Who's doing the research? How are they doing it? Who's funding it? What is the goal? etc etc Those are all pertinent questions. I don't think all academic research is inherently biased, but I've spent enough time in academia to know that there is a very heavy general left lean, so we need to be skeptical of politically charged "research".
Not immune to facts at all, but in this case the facts are anything but clear and easily discernible on an objective basis.
No, what you are doing is advocating a classic science denial position, which is that research is great when it tells you what you want to hear, but every time it doesn't, the researchers are biased. Just like the way conservatives are with MMGW or the way liberals are with GMO foods.
But conservatives are much worse of the two these days because they have internalized the belief that all academia and news media are systemically biased against them. This allows them to pick and choose. It's the ultimate in confirmation bias.
Conservatives can now construct their own reality based on using a tiny handful of examples to prove a trend involving hundred of millions of people, or engage in whatever bullshit fallacies serve to produce the conclusions they favor. There is simply no other logical conclusion to what you are saying here.
Research can be challenged BTW. You challenge it by critiquing its methodology, not by making a blanket assumption that all research which doesn't support your views is biased.