In fact it's quite simple, see answers below each of your questions:
1. Why three different shades of accents?
This is because you will need different states for buttons, nav bars and any other element that has subtle changes of color. While this is obviously most evident on hover, active or click button states, it's used in many other elements as well, see an example below where primary color has a darker hue for a second line of text:
which is based on this (see full page):
In long lists of complex information, consider using colored
subheaders to delineate content. An example of the use of color for
subheaders is in Android settings.
and of course, buttons: you will need to add hues and tints depending on the button state. These hues and tints are the 3 variations of the primary and secondary color, hence why you need them, see below:
Color version
Monochrome (light) version
2. The site https://www.materialpalette.com, once you pick your colors, also mentions "Divider color". What is that for?
Dividers are just that, dividers. You can see the different type of dividers here. As for divider color, Material recommends dividers to be 12% white for dark backgrounds, or 12% black for clear backgrounds. This is easily made by using RGBA colors. For example:
.divider{width:100%< height:1px; background:rgba(0,0,0,0.12)}
This way, by setting transparencies your divider will take not a set color, but your background's color + 12% of black (and same if it was 12% of white for dark backgrounds)
3. Materialpalette.com also has colors for text/icons, primary text, secondary text. What's the difference between "text/icons" and "primary text"?
OK, please don't take that site as a guideline, the official Material Design guidelines about color are very specific and leave no room for ambiguity . If you want to know about text, then your guideline should be Material Design: Text and Background Colors, not any other site, or you'll go crazy.
This being said, to answer your specific question: primary text color is 84% black on light backgrounds, and 84% white on dark backgrounds. There really isn't any other value, it's always the same. For secondary text, replace that 84% with 57%. And remember: these are just GUIDELINES, you can do whatever you want as long as it looks legible, but the fact is for Material Design, these values are static.
As for icon colors, they will always be white or black (or a very dark grey) depending on the color you chose for accent when in FAB (floating action buttons), but they get the primary text color if in regular backgrounds (light or dark). Here you may need to use your best judgment since it won't always fit fine, so don't get too attached to the "MUST BE", just make sure it works well and it's legible
