Different people use Photoshop in different ways, but here's some feedback with respect to curves vs levels:
Curves can do everything that levels can do, and more. If you understand curves, you really shouldn't need to use levels, and you certainly don't need to use both right next to each other, applied to the whole image. I cringe a little bit whenever I see people use multiple levels and/or curves adjustment layers in succession. Not only is it unnecessary, but you'll be throwing away color information. You'll see evidence of this in histogram combing and color banding. Of course, it may make sense to have several local curves adjustment layers that apply to isolated areas of an image (e.g. with masks), but it's generally a mistake to use multiple global curves (or levels) adjustment layers in succession.
It's probably not much of a problem for illustrations with flat colors, but you'll definitely notice it in gradients and photographs.
Yes, technically any set of consecutive lookup tables can be combined into a single LUT.
However, it's often easier to stack them and just makes sense. Plus, 16-bit color makes it technically possible.
Let's say that you're working with a color-sensitive piece in a color-correct pipeline. First adjustment layer should be to bring your pantone color card to look right. Then, the next adjustment right after that would be your overall grading pass, e.g "cold" or "vintage". Then, the next pass could be to "get those skin tones a little pinker". Much easier than working on a single global correction layer.
I agree with that. 16-bit color helps a lot (but doesn't completely eliminate the risk of throwing away data), and the approach you mentioned (correction vs grading) is a valid exception to my point - though if you're doing that, you likely understand what you're doing and the potential gotchas.
My point was more targeted toward the numerous tutorials I see where people use many such adjustment layers without real purpose other than to "play around" with the color, turning the image into a posterized mess (especially by stretching things too far one way, then essentially negating that adjustment with something else).
I love the idea of using a strong and somewhat arbitrary palette which is then tuned with "post-processing" layers. Instead of worrying about the exact shade of every color separately, you can achieve cohesive effects on the whole palette.
A lot of the pro level camcorders do this. For Sony it's called slog-2 and slog-3. Here's a video showing what it looks like: https://vimeo.com/110595557
Footage shot in logarithmic color space looks extremely dull and grey until you apply a LUT (lookup table) to it. I'm not sure if it's exactly what the Author is doing, but it is interesting to me when you see footage that almost appears black & white. Then just applying some magical math to it, all of these vivid colors pop out!
If you really want to learn about color you should start with a serious study of color theory. If you don't have the bandwidth to take a full course you should read the book "Interaction of Color" by Josef Albers.
Curves can do everything that levels can do, and more. If you understand curves, you really shouldn't need to use levels, and you certainly don't need to use both right next to each other, applied to the whole image. I cringe a little bit whenever I see people use multiple levels and/or curves adjustment layers in succession. Not only is it unnecessary, but you'll be throwing away color information. You'll see evidence of this in histogram combing and color banding. Of course, it may make sense to have several local curves adjustment layers that apply to isolated areas of an image (e.g. with masks), but it's generally a mistake to use multiple global curves (or levels) adjustment layers in succession.
It's probably not much of a problem for illustrations with flat colors, but you'll definitely notice it in gradients and photographs.