How to get a good clean newspaper scan?

its old school, but put a sheet of black paper behind the newspaper. A lot of the issues are caused by the fact that newsprint is thin poor quality paper. The scanner sees the front and the back of the page at the same time. Black paper matches the black ink and makes the "back" side of the newsprint disappear.

A quick fix I do is to compensate for the low quality paper and printing, by adjusting the 'white point' and 'black point', efectively increasing the overall tonal contrast of the image.

Looking at your first example, the lightest parts of the image -- unprinted paper -- are actually a light grey and the darkest -- 'black' ink -- are a mid grey. And there's "blotchiness" in the lightest areas caused by the texture of the paper. One accepts this when viewing the original on paper, but not when viewing on screen, or when reproduced in higher quality print.

In Photoshop (sorry, not familiar with the Gimp, but I'm sure it's a similar process), I add a Levels adjustment layer to avoid altering the original pixel values, and then use the white point and black point sliders to "clip" the lightest and darkest points of the image to pure white and black. There's are eyedropper tools in the dialog for setting these points by sampling from the image, but it's easier to adjust the sliders manually.

Depending on context, I might add the paper colour back in as a a very light grey underlying layer, then set the scan layer's blending mode to "Multiply". This is essential if the original is printed on coloured paper, like the Financial Times.

It's fairly tricky to scan halftone-dithered images from newspapers without a bit of background as to what's going on with the dots.

I suggest you have a look at this article.

For best quality, you basically choose the right resolution (the same as the print) and scan using a descreening filter (often available in your scanner software). You can then apply a despeckling filter (like median) to remove most of the remaining artefacts.