frequency and type of pulse for time domain reflectometer (TDR)
However, I could not find information about the frequency and type of pulse used by reflectometer.


Typical TDR instruments use a square pulse and measure the response only to one edge, so they are measuring the response to an approximation to the unit step function \$u(t)\$. 

The frequency range they're able to measure depends on the rise time of the stimulus pulse. Since the Fourier transform of even an ideal \$u(t)\$ has \$1/f\$ characteristic, and the feedlines to the device under test (DUT) typically have low-pass characteristics as well, it is quite challenging to measure very high frequencies with TDR, although commercial instruments from Tektronix or Keysight can reach the neighborhood of 20 GHz at least.

Another challenging in implementing TDR with an FPGA will be sampling the response with low noise and low jitter.