Digital output (FET) behavior
Note: The credit for this answer should go to @BrianDrummond, who provided an explanation in the comments.
So, it turns out that LEDs indeed can start glowing with merely microampere current. In fact, while testing them with various resistors in series I accidentally connected +24V to LED strip by touching battery with one hand and holding LED in the other. That was enough for entire 1A strip to glow.
The MOSFET 1NV04DP used in the digital output has leakage current 75uA @ 24V. More than enough for LEDs to light up.
Our solution to this problem was to add 47k resistors in parallel to each output.