... I was wondering which one of these palettes is really in control of
the color mode and how the file will print.
Neither of them, really, but the Swatches palette comes closer than the Color palette does.
The Document Color Mode command (from the File menu) is in control of whether a document maintains color definitions in RGB or CMYK.
Both the Swatch Options palette and the Color palette allow you to view and specify colors using "foreign" sliders (RGB or HSB sliders in a CMYK doc or CMYK sliders in an RGB doc), but before saving them they get converted into closest color consistent with the document's color mode, using the current document's working space Color Management Profile. So the foreign settings are for transient display only, regardless of whether you are setting them in the Swatch Options or the Color palette.
Grayscale has special treatment in that defining a color in Grayscale means that inside a CMYK document, it will print only on the Black plate (no rich blacks), and inside an RGB document it will translate to equal percentages of R/G/B. This makes Grayscale a nice way to represent black when objects may be moved back and forth between RGB and CMYK documents, or when defining a library of resources that may be used in either color mode. (For this reason, the default graph colors and the default black/white gradients are specified in Grayscale.)
However, global color and spot color definitions cannot be preserved in Grayscale mode!! So as soon as you make a Grayscale color be global, it gets converted first to black-only CMYK, and then if it is an RGB document, it gets converted to RGB under the current profile, which generally causes a shift to slightly unequal RGB values, and if converted from RGB back to CMYK, generally causes rich blacks to be generated.
Spot colors can be remembered in a "foreign" color model, but global process colors cannot be. So if, in a CMYK document, you use the RGB sliders to set the color values for a spot color, then the color will be remembered as RGB, and each time it prints those RGB colors will be converted on-the-fly to CMYK, using whatever color management is active at the time it is printed. If, however, it is a global process color, then the RGB values will be converted to CMYK at the time you click OK, and that's how they will be stored.
The Color palette will show the Tint slider if the current color is a global or spot color, regardless of which color mode that global color's values were set with.
For non-global colors, the Color palette will whenever possible show whichever set of sliders you last used, if the current color can be represented that way. So if you last used the Grayscale sliders, then if you pick a CMYK color that contains only black, it continues to show the Grayscale sliders even though the color is internally defined in CMYK. If you last used the RGB or HSB sliders, it continues to show the RGB or HSB sliders, even though the color is internally defined in CMYK. The color mode in the Color palette is not showing you how the color values are stored, it is showing you the sliders that you've been using recently.
The most reliable way to tell how a color is stored is to select an object painted with that color, and open up the Document Info palette.
Global colors mean that the color is "linked" to the swatch, so that any time you redefine the swatch using Swatch Options (or by drag-and-drop or Merge Swatches), all objects that were painted with that global color will pick up the new definition. You cannot directly adjust global colors in the Color palette except by changing their tints. When you pick any non-Tint set of sliders, you are breaking the link between the object and the swatch so that it no longer uses that global color.
Non-global colors are not linked to the swatch. When you select an object and click on a non-global swatch, the object has no memory of which swatch was used to paint it. So redefining the swatch will not have any effect on colors that were painted with it. It is more using the swatch as a starting point for a color mix, like a painter's palette.
Global colors are best used when you have multiple objects painted with the same color, and you expect that you may want to change the illustration at some point so that everything that was painted Color A becomes Color B. Document-wide changes are much easier when using global colors.
Non-global colors are best used in more painterly contexts, where you may have hundreds of shades of very similar colors, and you frequently want to fine tune the hues on one object without affecting other objects that may have initially been the same color. In this situation you may set up your Swatches palette like blobs of your main paint colors, and use the Color palette to do the fine tuning. Global colors would be less convenient in this context, because you would not have immediate quick access to the color channels, and every time you wanted to make two formerly identical colors diverge, you would have to break the global link.
As far as grayscale colors go, if you are working strictly in a CMYK document, there is really no difference in how a document prints between setting a color using the Grayscale sliders and setting it using the K slider of a CMYK color. They will both print using the specified percentages of Black only. It is only going through RGB that "turns on" rich blacks in colors that were not explicitly specified to have CMY components. So there is no downside to using global colors if you want to be able to globally change the percentage of black in some set of objects (say converting all colors using 15% black to using 20% black.) But if you are going to be in a mixed web/print workflow where you frequently transfer objects between CMYK documents and RGB documents and then back again, you will probably want to leave the colors as non-global grayscale, so that the cross-document transfers will not introduce CMY components into your blacks.