The rotation rate varies. It's typically about 3 RPM in precipitation mode. It scans 14 elevations per volume, so 14/3 = ~4.7 minutes, which is why most volumes scans take "about 4-5 minutes". In clear air mode it only scans 5 of the lowest elevations per volume and it takes about 10 minutes, so that's about 2 RPM.
It scans one elevation per rotation. So, it starts at 0.5 deg and rotates 360 deg. Then the antenna is raised to 0.9 deg (for precip mode) and rotates another 360 deg. And so on until it reaches the highest elevation angle.
I was inside the radome at the Melbourne, FL NEXRAD while it was down for maintenance. It's a massive antenna - 28 ft diameter! At the time, the Melbourne radar was about 65 ft above ground (now it's 90 ft) and climbing the open stairs to the top with a 25-mph sea breeze was a bit unnerving.
I've been wondering about the TV station radars too. They all used smaller 5-cm (C-band) radars and those that had them used to tout them as something unique. But I can't recall a TV station mentioning their own radar in many years. All the DC TV stations show NEXRAD, although some seem to indicate they are using the Dulles, National, and Baltimore TDWRs, which I doubt, but they do show the TDWR locations on their map as if they're integrating the data. Oh, and they often show the "fake" radar sweeps zipping around the radars.
Enterprise also made radars for TV stations. Before I left Florida in 2015, the 45th Weather Squadron at Cape Canaveral purchased and installed a new C-band Doppler radar that replaced an old WSR-74C (from 1974) at Patrick AFB. The new radar was positioned west of the Space Center to be able to better see the sea breeze since Doppler "sees" wind perpendicular to the radar and the old Patrick AFB radar was parallel to the coast and south of the Space Center, as is the NEXRAD at Melbourne. The new radar gave us excellent depictions of the sea breeze and outflow boundaries.
There's a company called Climavision (
https://climavision.com/) that is deploying 3-cm (X-band) radars as low-altitude gap fillers for the NEXRADs. They started installing them around Charlotte, NC last spring and have continued to expand their network (see the "press releases" section of their website). Their business model is to build and maintain the network and charge users for the data.