Regarding the PNG vs JPG thing: It's purely anecdotal of course, but I've experienced instances where signature sized (500x100) PNGs without transparency end up being smaller than sufficiently high-quality JPGs. That's always been with very simplistic cartoon graphics, though it doesn't have to be as low as 8 colors. However if you're saving an image with enough colors - evidently not a very high limit judging from your example Remi - you should roll with a high quality jpg.
A general tip that always works is "use the save for web tool and compare both formats before deciding". That way you'll always end up with the slimmest possible file, and if your editing software has such a slider, be able to tweak the Jpg loss levels to your liking. I'm fairly sure that tool exists in both Paint.Net and GIMP from what I've heard, so you shouldn't need photoshop (what I use) for that.