Good questions. I’m no expert but here is my thinking about LAB:
a) It is properly a color model rather than a color space. The space it covers is the full CIE visual color range. The history is a bit complex and the ultimate goal of the system was not quite achieved but basically the idea was that a one unit change in any of Lab for any color should have a comparable effect as for some other color. Also, it divorces luminance completely from color, which helps avoid unwanted color shifts.
b) that avoidance of unexpected color shifts is one of the main reasons for my using Lab (which by the way should properly be written Lab* so as to avoid confusion with some older Lab systems). Sometimes I find it easier to just think in Lab terms.
c) I tend to use Lab for images that “need more work”. For an almost perfectly exposed and lit image the RGB tools are more than adequate. Lab comes into play if more drastic editing is called for.
d) I do mix and match Lab and RGB tools but but only if necessary as the choices of what to alter for what end get very complex.
e) same for chrominance, best used alone in most cases but don’t hesitate to try mixing if it seems necessary.
f) using the White balance can accomplish similar things to Lab but it is easy to overdo things in White Balance. White Balance seeks to keep things related to “true” white as defined by black-body radiation (check out Wikipedia if this is new to you) so it is great if the whites in your image seem tinted. Again, there is no automatic reason to prefer either tool over the other, just whichever accomplishes the goal most easily for you.
g) I often work with the L curve in Lab, rather rarely with the others but they are all very useful for dealing with specific problems.
I’m sure others who have a stronger knowledge of image processing may have other thoughts and I am happy to receive corrections but this is my best set of answers to your questions.
Louis