When I move cursor on an anchor point, Illustrator shows me the coordinates of the anchor. But how can I find the [x, y] coordinates of the handles (in red circles)?
The easiest way I know to get Illustrator to display the actual X,Y of the handle is to use InkScribe from AstuteGraphics.com combined with the Info Panel.
Using InkScribe you can highlight a handle and then see its exact x,y coordinates in the info panel.
You can simply use the Pen Tool in Illustrator combined with Smart Guides as well.
If you hover the pen tool (I have precise cursors on) over a handle until the smart guide label is shown, the info panel will read the coordinates of the cursor, which is directly above the handle.
The difference between these two is the "stickiness" when using InkScribe. You don't have to be as exact with InkScribe, the handle is easier to target. Using the Pen Tool you have to hover over the exact screen pixels which initiates the smart guide label, then not move the cursor until you've read the info Panel.
Both work, one is merely easier.
The only thing I can find are relative (spherical) coordinates with respect to the anchor point. It's not helpful on it's own, but maybe you can use it in a script.
Window -> Infoor cmd+8).
Direct Selection Tool.
Infopanel will show the distance
Dand the angle with respect to the anchor point:
Combining that knowledge with the coordinates of the anchor point you can calculate the coordinates of the handles using basic trigonometry.