1)I see. Well, I believe that the reason here is the export option of the image artist used was a lossy compression format(image that irreversibly reduces the original quality of the image) for example PNG-8. That caused the quality loss. I forgot to ask you this, what was the format the image artist used for export?
2)Good that you didn't use the Construct 2 Image Editor for complicated images since it is not designed that way. I think Paint.Net is good as long as it has the option to export to different known image formats but I recommend Photoshop, Illustrator or other known and high reputation image editors if you already own one.
3)Good you shouldn't re-size it on Construct 2 since it makes it worse.
4)Good since it reduces quality if it is stretched.
5)In question 1 I asked you if your image was a whole image and you said yes. Well, in game making you shouldn't make a background that way. You shouldn't make your game parts as whole images, instead you have to use tiles, sprites(by use of sprite sheets - optional) or 9-patch. Because having a big image in your game is very inefficient and can make a game non-playable:
Please read this:
https://www.scirra.com/blog/112/remembe ... our-memory
6)Since you are making a pixel art. I suggest re-sizing your window size to 640px*360px so that it will be easier in memory use but you can re-size your layout all you want depending how big your map/game is. And yes, set your sampling from "linear" to "point" to prevent blurs and also set pixel rounding to "On".
*Export your image to PNG-32 to avoid quality loss. (Blurriness Solution)
*For performance and practical/professional game making. Instead of a whole image background, use sprites, tile background, tile maps and 9-patches for your game.
*Set Sampling from "Linear" to "Point". (Blurriness Solution)
*Set Pixel Rounding to "On". (Blurriness Solution)
*Set Window Size to 640px * 360px.
That should fix everything. I suggest you read the manual too to learn good game making practices: