The Illustrator width tool can be found in the tools box.
They are generally available when you see the stroke options for a path and can be located in numerous places throughout the application
They can be used with paths, type etc, anything that has an accessible stroke. Select the path, select the preset from the stroke drop down and apply and modify stroke size etc. You can use them with a variety of tools such as the pencil tool. They can be found in the properties panel etc again associated with the stroke settings. Some features do not allow them to be used.
Yes, that is it's task, as long as there is a stroke associated with a path, you can use the tool. It doesn't need to a preset applied either, it can be used with a basic stroke. Go to any point on the stroke with the tool and hover over the stroke and you will see a dot appear as you move around and if there is no existing point, the cursor will change to a '+' and you will see a dot appear to indicate that you can add a point there and modify the stroke. You can then drag the tool out from the center of the stroke and change the width at that localized position. You can add as many points as you wish, you can also delete them by selecting them.
As you drag in and out on the stroke, you will notice that the stroke changes symmetrically, side 1 and 2 are the same. Hold down the ALT / Option key then you will notice one side will increase or decrease but the other side is untouched
You can select an existing point on the stroke and drag that point to a different location but there is a barrier, another existing point on the stroke. As you move the point between those two barriers, you will notice the curve of the stroke changing
You will hit a limit as you extend the stroke, on my current version it suddenly stops at 500pt. So if you select a point with the tool and drag it outwards, it will hit 500pt and that will be that. Perhaps that will be updated in future versions.
Select to 'uniform' (in the drop down) and set the stroke weight to 2pt or 5pt etc
You can also set other stroke properties such as dashed and they will change as the tool modifies the stroke. You can use all the usual settings such as cap / align etc as well as use any combination of dash and gap.
You can combine them with brushes. Select a path, modify it using the tool and then go to the brushes panel and apply a brush and you will see that changes as you modify the stroke
Select a path, add a profile or manipulate the stroke with the tool and set the color for the stroke. Go to the appearance panel (window menu) and then right side menu and duplicate stroke entry. Change the stroke weight as well as color and also you can edit the different strokes independently. To edit them, select the entry in the appearance panel and then use the tool and the currently selected stroke will be changed and all the others will be ignored.
You can use the tool with the repeat feature. Select a path and apply a preset to the design and then select the object menu and repeat and radial. Double click one of the instances and then change the preset using the tool