I've also tried to buffer the signal,
With what? Ultimately this is the answer. A buffer at the drive end and one at each of the receivers would be best.
TLC's are made for daisy chaining so you would expect that there are ways to link a great many
There are but not at the distances you ate expecting them to go at.
The library uses pin 13 for the clock signal,
That is because of the hardware inside the processor attached to that pin.
can it be that the led and resistor on this pin have an influence,
They will not help as they will take some of the drive capacity away. You could try removing the resistor.
However how farther down the chain i measure how larger the amplitude becomes (which i think is strange).
Yes strange but not un understandable. You are measuring at various points in a distributed network of LCR (inductors capacitors and resistors) and the inductance effect will sometimes show this.
with 6, 5 will work, but with 12, 8 or 9 will work.
That sounds like a transmission line effect, where signals are reflecting off the impedance discontinuities at the end of the line. The answer to that is to match the impedance of the line with the correct size terminating resistor.
There is a nice animation of that here:-