What type of onboard radars were used in the Surveyors and how did they work?
Surveyor measured the rate of descent with radar by using Doppler shift.
From the book Planetary Landers and Entry Probes:
The surveyor lander used a pulsed radar to generate an altitude reference at 100 km, a separate RADVS (Radar Altimeter and Doppler Velocity Sensor) turned on, using a four beam frequency modulated continuous wave technique (Figure 5.4). A central beam at 12.9GHz was used to measure the altitude, while the three outer beams at 13.3 GHz were used to determine the three components of the velocity. To accommodate the large variation in range (and therefore signal strength) amplifiers provided 40, 65, or 90 dB of gain.