Do Echo Tops always register as low when very close to the radar site (including when a storm passes over the radar site), or is what RadarScope shows in the Echo Tops option always accurate regardless of radar location?
Yes, they do estimate too low near the radar site. The maximum elevation angle scanned by the WSR-88D is 19.5 degrees above horizontal, and very close to the radar site, that isn't very high in terms of height above ground level. If you want to get an estimate for what the echo tops are near a radar site, it's possible (if the radars aren't too far apart) to use an adjacent 88D and look at their Echo Top product in the vicinity of the the radar you were looking at.
The height of the beam is very important to know when looking at all radar products. Not only can echo tops read low near the radar, but things far away are impacted as well. For example, in Paris Texas the lowest scan from the KFWS radar is 15,000 feet above ground - if you saw rotation over that area on radar at the lowest scan, you are really looking at mid level rotation. You can use the RadarScope distance tool to tell you the beam height if you drag from the blue dot at the radar location.