So you don't want to read the whole, long story? In a nutshell: 

-the console displayed images with wrong colors (gradient from green to blue) in the first case. Only in PAL-N the colors were right. That's strange since the console is probably NTSC unless it was modded to work in PAL-N. The image showed the gradient afgain when set to PAL-M
-It displays checkered monochrome images in my automatic tri-system TV 
-It displays monochrome images in my parents' TV, but PAL-M shows a clear monochrome image (no checkered pattern).

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I bought a SNES and hooked it to my boyfriend's Samsung 50" Smart TV. It displayed images with an horizontal gradient from green to blue. Later I discovered it was caused by the "Auto" TV system setting. So I tested it with the three available systems (all modern TVs sold here are at least tri-system). The green-to-blue gradient was caused by PAL-M. NTSC displayed black and white. The only system that displayed colour correctly was PAL-N. That's kinda strange because Ni****do never had official distribution in Argentina. And AFAIK all SNES sold here were American NTSC systems. 

Then I tried it on my 32" Philips LED TV. All I get is a checkered monochrome output. The TV is an automatic tri-system (PAL-N, NTSC and PAL-M) TV. It's ot possible to switch manually between TV system. So that's the end of the line. It doesn't seem to be a solution for this one. 

Then I hooked it up to my parents' bedroom 32-inch Panasonic Viera LED TV. Much to my surprise, it started in colour. but then it began displaying monochrome output! (I tested it with NBA Jam, and it switched from color to monochrome when the Iguana Entertainment logo appeared) I switched between the three available systems and the Auto setting to no avail. But there are differences between all the settings: PAL-M (the same system that caused a gradient on my man's TV) gave me a clear, non-checkered monochrome image. 

Is this common? What causes this problem? Someone suggested me to switch all image enhancement off and switch to Game Mode, but that didn't work.

If you have the tools to open the console, please do so and take a picture of what you find inside.

If you find a mess of wires, than it may be a conversion mod of some sort. In which case it is difficult to tell what is going on without somewhat expensive electronics test equipment and knowledge of analog video circuits.

If not, you should find a little green trimmer capacitor that you might be able to tweak to bring the color back. I had a friend who had an NES without color due to the trimmer cap being off.

I'd also suggest trying things like the s-video and RGB outputs if you can.