Regarding JPEG compression, there used to be a site that would make a heatmap of JPEG-induced noise for a given image, so you could see hotspots where it looked like it had been compressed twice (i.e. manipulated).
It's noteworthy that the first image in TFA doesn't have obvious signs of tampering in that tool
Yeah, that was it. They shut it down but still have an info page on it: http://www.errorlevelanalysis.com/