Should a traffic light be mounted on the RIGHT side of the road when turning right in
 

I got a red-light ticket because I didn't stop on North & IL 59 S in West Chicago. I turned right but did NOT see a traffic light on the RIGHT side of the road. The only traffic light I saw is circled in red in the image below. But I thought it was for traffic that goes straight. I've never seen a road without a traffic light on the RIGHT side of the road or across the road (circled in green). It's very confusing. Should not there be a traffic light on the RIGHT side of the road for turning right? Below are two images of the same intersection. Can I contest such situation?
https://cimg6.ibsrv.net/gimg/hondafo...e9a0043a44.png
https://cimg4.ibsrv.net/gimg/hondafo...cf9b1f43e3.png

Not legal advice
 

If it were me and I described the incident like you did and omitted critical information, I would pay the $100. fine and move along.

The answer to one of your questions can be found in 625 ILCS 7. Both of your photos clearly show a solid white stop line painted on the turn lane.

You have 7 days to file if you want to contest the ticket. Should you?

Hard to say because you failed to specify whether your ticket was initiated by a traffic camera or by the gendarme.

If it was a camera you would need to see the camera photo. If it was the gendarme you might be able to use your photo with the green not a circle to claim the stop light was inoperative.
See attached photo you posted edited to include a blue circle around the subject light.
https://cimg6.ibsrv.net/gimg/hondafo...017b68817.jpeg

They like your money.

The first image is the real image from traffic camera. It was indeed red as you can see in the image. The second image (the image you used to circle in blue) is NOT real image, it was taken from google maps just to show that there is no traffic light on the RIGHT side of the road.
So, like I said, I didn't pay attention to the red on the LEFT traffic light (circled in red) since I thought it was for the straight traffic and not for the cars that turn right. There should be a traffic light mounted on the RIGHT side of the road for the cars that turn right, correct? I've never seen a road that does not have a traffic light on the RIGHT side of the road for cars that turn right.

Reality
 

“There should be a traffic light mounted on the RIGHT side of the road for the cars that turn right, correct?”

Incorrect.