You have asked two questions.
Who designed the props?
I have no idea.
What inspired the circular computer screens?
This I can answer with some small degree of first hand knowledge. For this, you need to know a little about computer and electronics history.
A CRT is a Cathode Ray Tube. It works by sending a stream of electrons out, interrupted for ‘off’ pixels on the display. The radiation is directed by pairs of electromagnets, one pair to deflect the beam horizontally, and one pair to deflect the beam vertically. You can read more over at howstuffworks.com.
Very old equipment, such as an oscilloscope (link to Google image search) had an even coat of phosphorus on the display, and you could actually see the beam tracing back and forth (which was the point). Adjust the frequency low enough and watch for hours of fun. (Don’t be distracted by the square ‘lines’ — they were painted on the glass.)
The design was rather primitive, and a round screen was very much easier both to manufacture and to program for the bounds of the beam. Google finds pics of vintage TVs easily. The edges are rounded, not because of the signal encoding (the TV signal has always been for a rectangular image, IIRC), but because the image at the corners was either missing (cropped due to the bounds on the beam) or not very well defined (and hence fuzzy, and fuzzy is bad marketing).
More modern CRTs, like the stuff you grew up with in the 70s and 80s actually swept the beam much further than the (square) display, and had hardware to control turning the beam off past the edges, and a significantly more complex hardware for controlling the beam to strike the RGB phosphors in color TVs. (Get yourself an old EGA/VGA hardware manual if you want some interesting technical reading — you can control the bounds directly with software! :-)
Now, the idea of round displays was not new even with Star Trek. You can see round display screens in older sci-fi like Flash Gordon. Here’s a pic I found of an old FG comic featuring a round view screen. Looks a lot like an oscilloscope, doesn’t it? Round displays even featured in TV episodes along with the bleach-bottle ray guns.
During the late 40s and through the 50s, when people thought of hi-tech stuff, they thought of these cool devices that showed a picture. They thought of early televisions, which in the 60s still were new, high-tech stuff operated by nerdy science-y people.
Even today round is sleek and high-tech. I bet you can easily find many sci-fi movies featuring pretty, round/curved, holographic/see-through/awesome futuristic displays, with no square corners, all on your own.
Moving, drifty, waving lines
Simply, money. Oh, and corporate aesthetics. Psychadelic, cool stuff doesn't look like the text-graphics computer displays in the Andromeda Strain. .
Star Trek (TOS) was shot on an exceedingly tight budget given what they were doing. Masking part of an image to place another above it was a time-consuming technique. Doing things like lining up a spaceship against a planet was an expensive task. Why spend time and money to display something that we would consider ‘real’ that had no effect on the plot? It was much easier to simply put up a pretty color wash that got perfectly cropped by the masking.
Even if Spock had to look into the display and ‘read’ something, it wasn't too far-fetched. A lot of old optical equipment could only properly display things to people directly in front of it. Those in the audience’s position would not see anything more than colored light. I suppose that on Star Trek the displays were just more advanced so you got a lot of colors. That or it was a screensaver, LOL.
So, to finish, the main points are:
- It wasn't really new, just pretty and more futuristic versions of stuff we already had
- It was cheap and easy to produce that way