JC, the key to a current sink LED driver is to have enough 'overhead' on the current sink pass element to maintain regulation.
Inside it is a linear regulator, as such it must have overhead to operate in its linear region.
Below is a graph of Vout vs Iout of the current sink in TLC5926.
For instance, at 100mA sink you need 0.5V minimum at the IC current sink pin.
This would relate to the sum of the LED Vf forward voltages in a series LED string, subtracted from the power source at the top LED anode.
Second question, yes, the voltage appearing at the IC sink output can be up to 17V.
Operation will be limited by the total power dissipated in the IC.
Design guidelines would be to assure at least 0.5V at the IC sink pins with worse case maximum LED VF and worse case minimum supply voltage (<=17V).
Third question, NO.
The IC sinks current to ground, sourced from the LEDS, so the IC does not provide the LED current, just provides a path to ground.
IDD current is the current INTO the VDD pin and out of GND.
OUT0-15 current comes INTO the OUTx pin and out of GND.
A side note.
IC power dissipation will be based on 2 things.
1: IDD x VDD for the IC quiescent operating power
2: Sum of the Isink x Voutx, sum of all the individual OUTx pins sink current and their voltage when sinking (0.5V or greater)
Does this help?