Did Kenny from "Shut up And Dance" in Black Mirror really do what they say he did?

Yes, he was watching child pornography. This twist is confirmed in an interview with Alex Lawther who portrayed Kenny and adjusted his performance accordingly:

Interviewer: How do you maintain that level of intensity?

Alex: [The director] James Watkins kept giving me gentle nudges, reminding me how high the stakes were for Kenny. The audience might be thinking, "Oh, come on. You haven't done anything that wrong." And then we find out exactly what he has done.

Interviewer: "Shut Up and Dance" seems like it's about a boy driven to desperation because he doesn't want his friends and family to see a video of him masturbating to online porn—which would definitely be embarrassing, but maybe not so much that you'd rob a bank or beat a man to death to prevent anyone from seeing it. And then, at the end of the episode, we learn that Kenny was actually watching child pornography. How did such a disturbing twist inform your approach to the character?

Alex: For Kenny, it's life or death, really. And that drives him, in spite of his terror, to carry out the tasks he needs to. For me — considering each event [from Kenny's perspective] as it came in the shooting order, as life or death—the stakes were always incredibly high.

And you could spot a few subtle hints during the episode. Kenny seems way too horrified throughout that he was caught and goes to enormous lengths to conceal it. Also, his interaction with the little girl at the restaurant in the beginning of the episode reads very differently upon a rewatch. Also note that the hackers simply reveal the truth about everyone in the end; they didn't need to make anything up. In the Black Mirror-verse, technology isn't the problem; it's humanity's flaws, that are inherent in it. Technology simply exacerbates them.

If you jump to the scene where he meets the guy in the woods to fight he says that "I just looked at some photos" and the other guys says "Yeah, yeah, I just looked at pictures too."

But then the other guys he asks "How young were they?" and Kenny just shakes his head and starts sobbing and they guys says "Yeah. Well me too."

So this strongly implies they were very young.

And then he gets a call from his mum: "What did you do Kenny? They're saying it's kids! That you've been looking at kids!"

So wanking off to pictures of kids... nothing "regular" about that.

I'm gonna bend the rules a bit and approach this question from a different angle. Does it matter? How did they show what he was watching?

Since they recorded him with his own webcam, how would they know what he was watching? Sure, they can also record his screen but then they would have to compose both streamings in a single video, and that would be suspicious and/or easily deniable. How do you prove he was actually watching that and is not something you just added in editing?

Sure, some neighbours would actually believe anything they see and may not listen to reason or explanation. We know neighbours love a gossip, and ostracising feels so good and self-rightous...

Sure, his ISP could share his network activity if solicited by the police[citation needed] (I don't know the laws regarding that in the actual UK, much less in the fictional UK setting where the episode is set)

It is supossed to be a big twist at the end:

Alex: [The director] James Watkins kept giving me gentle nudges, reminding me how high the stakes were for Kenny. The audience might be thinking, "Oh, come on. You haven't done anything that wrong." And then we find out exactly what he has done.

But as shown in the episode it still seemed weak. They move the stakes of the shame of the masturbation act to the shame (and, actually, crime) of what he was watching. And it that moment, for a techie like me, my suspension of disbelief went out the window. The important question remains unanswered, which is: how did they show what he was watching? You could say that, as an spectator I was one of those rare skeptic neighbour who needed stronger proof to believe he actually did that. It is shown as a very dramatic situation but that could actually just had been overreaction of a teenager and his mother.

At the end, for better or worse, it doesn't matter if you did it. It matters if they can reasonably prove it.

DISCLAIMER: Of course it matters if someone is a pedophile, pedophilia indulging that disorder (in the form of watching such content and supporting that industry) is a heinous act which I do not condone nor tolerate. I'm talking about the stakes and rules that the episode presents, because the episode could work pretty much the same with any other apparently-shameful-activity-that-turns-out-it-was-actually-something-despicable-and-illegal.

I think it's a little ridiculous to use Kenny's interactions with children at the restaurant as evidence for why he is a pedophile. I rewatched those scenes a few times, taking into consideration what others said about how he interacts with the girl and how he touches the crayons. I think the way he interacted with the little girl was perfectly normal. He was looking in their direction as they were leaving because he's a bus boy and that would be his next table. He spoke to her kindly and got down on her level. He smiled and waved. That doesn't strike me as creepy, that strikes me as mannerisms for a boy who has a younger sister and generally likes kids. Moreover, I don't think he was touching the crayons in a strange way. His fingers lingered on a few crayons while he was looking at what a child drew on their place mat, which was a burning building.

Yes, Kenny was a pedophile. It also makes sense because the hackers only blackmailed people who "deserved" it: Hector cheated on his wife, the man Kenny had to fight to death was also a pedophile, the woman you see at the end was racist, etc. I don't think they would make a teenage boy who masturbated rob a bank and kill a man.

I didn't want to think Kenny was, but the interaction with him at the beginning in the restaurant with the young girl kind of swayed me coupled with the only people forced to fight to the death were two accused of pedophilia.

Yes, he did. The most clear indicator was not the encounter with the little girl in the restaurant. Rather, after a few scenes later, it was the way he touches the crayons when cleaning a table after kids. Remember, the "Kids Menu"?

I think He gives it away as the episode goes on, he goes from saying they videoed him masturbating to he was just looking at some pictures to then A picture. Add that to what the man suggests when they have to fight to the death and his reaction, he clearly doesn't want to accept what he's done but KNOWS they know. They'll have all his hard drive info (as Hector said to Kenny in the car) and I think actually it's after that point he goes from worrying about a video of him masturbating to what he was masturbating to! The realisation of what's going to get leaked makes him into a far more desperate individual. So yeh, it's not only given away by what the man in the woods says or what his mom says when he calls but mainly from him shifting focus on what they have on him.

The answer was at the end when the police came for him. His mom called him and if you listen carefully, the mom's saying, "Is it true?! Were you looking at pictures of kids / They say they're just kids!"