JPEGs are thoroughly adequate. They were designed to be print sufficient and at 8 bits they are. The work done by the Joint Photographic Experts Group is frankly brilliant and the JPEG standard is a technological triumph. And to answer your first question, no the eye can not determine that the colors are limited in a JPEG.
Funny thing about human vision, it's self correcting whether you like it or not. I had a delightful experience a couple years ago when I had a cataract removed from my left eye. The new lens they put in my left eye gave it 20/20 vision -- no longer needed corrective glasses. About a week into my recovery I was seeing pretty well and I removed the eye patch they gave me. I then removed the left lens from my eye glasses and when I put them on I got double vision. My two eyes were seeing everything two different sizes. It was bad enough that I was struggling to get around. But I knew what to expect and what to do. I went for a walk in the neighborhood. As I walked my brain went to work reconciling the two different size images. After about 30 minutes it was doing pretty good. After an hour the problem had nearly been solved. When I got home after an hour and a half walk the problem was gone -- human vision is self correcting and that includes for color.
So we are able to discriminate very subtle color differences when we can see them side by side. But our ability to remember a color is less than abysmal. Test your ability to discriminate colors here:
Free Online Color Challenge and Hue Test; X-Rite and test your ability to remember colors here:
Color Game — How Well Can You Remember Colors?
What this means is that the colors in a photo have to be pretty seriously off for the average person to see a problem. That's why most people are happy with auto WB. It's never right but it's usually close enough and human self correcting vision picks up the difference. We don't normally look at a JPEG alongside a 16 bit TIFF of the same image and try to spot the color differences. And if we're only looking at the JPEG by itself then we're not in color discrimination mode we're in color memory mode -- less than abysmal.