@aarg I believe you meant 'sink' not 'supply' 60ma..
It is definitely not wrong to use "supply". Whether sinking or sourcing you are still supplying current.
Yes it is.
Okay, then... "provide" 60mA.
The TLC5940 has open drain detection that shuts down the output when it falls below .3V. By touching the outputs you are probably discharging yourself into the output and allowing normal operation.