The copyright owners don't care.

 

Feel free to hmu if you want a custom copyright-free fair use profile picture

 

If the picture is "official" or otherwise incorporating "sufficient creative effort" by someone else, yes they have (at least some) of the copyright on it
So the next question is, how many reasonably provable* damages did your likely unauthorized duplication result in?

* being a civil case, it's "reasonably" and not "beyond any reasonable doubt"

Next, you said copyright, but when you talk about characters (rather than limiting the discussion to their representations) it's more of trademark territory - as you may have heard there's no "do nothing" option against those violations, but you can avoid violating it in the first place by avoiding any confusion that you may be its creator/etc (ie there's a difference between putting Pikachu in your 100% free video game, in an iPhone cover sold for profit that clearly acknowledges being unofficial, and in a space where most uneducated people can quickly tell is used by anyone as "my favorite image")

 

What would happen if you port scanned or successfully hacked a foreign server?

 

Under the strictest senses of the term then yeah you would be using a copyrighted (possibly also trademarked*) work without permission, and I don't particularly see a fair use/fair dealing argument. However if they dragged you into court they would also want to be in a position to prove losses or that it provides unfair competition for their work.

At the same time even a company as stupid as Nintendo will probably recognise the value in having a character plastered around the joint as free advertising of a sort.

*trademarks have a concept called genericide where if a holder does not enforce their trademark then it becomes void. I doubt simple forum use would come close to bothering this though.

Probably the two main scenarios in which something would be done would be a) something like Scientology where they go after leakers of their "works" with copyright a lot of the time if you chose to make it such a thing, and b) if they thought association with a site/group was damaging the brand and just wanted shot of it. We have seen a few instances of b) before among the various political video circles, though several of those had other factors at play as well.

 

Fair use with pictures is a bit of a mess and something I had to become very familiar due to my old job. That being said, unless someone is going out of their way to find each person using their pictures and tracking them down, sending a C&D, etc.. In short, no is going to copyright strike an avatar because that’s a lot work with no pay off

 

Going through the usual four factors thing ( https://fairuse.stanford.edu/overview/fair-use/four-factors/ ) I reckon I could make a case and dodge a de minimis argument (typical avatar shots could do some heart of the work type stuff), especially if it is combined with a notable person (think forum mod/face of the site rather than libel law) on a site/person in some disrepute (here I presume they would go for the piracy angle, though I reckon I could also do a standing in the community argument for something like Kotaku).