Displaying 24fps content on a 60fps screen repeats frames. If you repeat half the frames of the 25fps stream twice, and alternating half of the frames three times, then you get 60 frames per second, with a stream looking like:
AABBBCCDDDEEFFF...
This isn't perfect, since really each frame should be on the screen for exactly the same amount of time, which is why 120hz TVs came about -- 120 is evenly divisible by each of 60, 30, and 24. But it's also pretty much good enough, or else there'd be a lot more complaining about watching movies on computers or other 60hz devices.
A 30hz screen makes that harder, since you're going to have to repeat every fourth frame once, so the unevenness could be more noticeable since those frames are now twice as long instead of 50% longer. But I haven't tried one to tell you just how noticeable it would be.
Native 60fps content (sports is the most common one I know of) would require throwing away half the frames, but at least the remaining frames would all be shown for the same amount of time.
NTSC "video" content (stuff shot at 30 interlaced frames per second, so 60 fields per second, stuff like old TV shows) would work alright. You wouldn't be able to take advantage of deinterlacing methods that try to reconstruct 60 full frames, but it wouldn't be a huge loss.
Actually in this case it would work fine. The Seiki's panel is actually 120hz, the 30hz limit is on the input side. Send it a 24hz HDMI signal and it can easily match that up to the panel's actual 120hz refresh rate by simply showing each input frame for 5 panel refreshes instead of the 4 refreshes it displays the 30hz input at. The Seiki can also take 1080p input at 60hz, which would be a reasonable fallback for gaming. Run 4k 30hz for desktop usage, and 1080p 60hz for gaming.
That's one thing I never understood. With CRT's, you still had discrete pixels (either clusters of 3 RGB dots, or 3 RGB bars). But no matter the resolution (within the CRT's limits), nothing looked blotchy. Whereas on an LCD, you can always tell when it isn't running in a native resolution (or even divisor). So what is the difference? Other than a CRT would naturally do analog interpolation, couldn't an LCD or graphics adapter be programmed to simulate the same effect?
I am not a hardware guy, but from my understanding a CRT uses an electron gun a distance from the screen to cause a matrix of phosphors to emit light. Hence the depth of CRTs. The electron gun can repeatedly scans from top to bottom, left to right (or whatever direction).
It can be adjusted to make any number of vertical "lines" as it sweeps over the phosphor matrix, but the matrix has a fixed dot-pitch. Because an electron beam is being shot at the matrix from a distance, there's a certain degree of bleed-over to neighboring phosphors. Firing electrons at the phosphors at coordinate 100,200 means a little bit of energy is also received by 99,199 and other neighboring "virtual pixels."
In other words, while the phosphor matrix was just as rigid as the pixel matrix of an LCD, the power source that caused light to be emitted was not perfectly matched 1:1 with the matrix. There isn't an electron gun for each pixel, there is a single electron gun for the whole display.
By comparison, each pixel in an LCD monitor (putting aside the three internal pixels for colors) is an individual mask over a uniformly white back-light. The panel adjusts the opacity of the mask to allow more or less of the white backlight through. The mask over a backlight is what gives LCDs a challenge with viewing angle (because there's a non-zero gap between the light source and the mask and the mask itself has a non-zero depth).
Finally, as I understand it, a plasma television is a bit like both. It's a matrix of phosphors like a CRT, but with individual power sources for each pixel like an LCD. Because light is emitted at the surface, plasmas tend to have very good viewing angles.
OLED then is, in my opinion at least, a spiritual successor to plasma. OLED emits light at its surface like an LCD or plasma, but uses no mask layer over a backlight, like a CRT or plasma.
(Take all of the above with a grain of salt because I'm just a layperson when it comes to hardware.)
The hardware side is where things start getting out of my area of knowledge, but my understanding is that yes, CRTs had a native upper bound on their resolution which had to do with the dot pitch of the display. I would guess that the way this worked in practice is that for lower resolutions they were effectively partially lighting up multiple phosphors per pixel on each pass? I imagine the difference between this an an LCD is that the amount of time a single pixel was lit by the beam on a CRT is far, far lower than the refresh rate of an LCD.
I believe there were monitors that went both ways with this, not just letting you run lower than native but also higher than native (I had a 17" CRT that claimed to support 1600x1200, but not without a lot of blurriness, for instance).
edit: man, it's really hard to find specific info about how exactly that worked online these days
AABBBCCDDDEEFFF...
This isn't perfect, since really each frame should be on the screen for exactly the same amount of time, which is why 120hz TVs came about -- 120 is evenly divisible by each of 60, 30, and 24. But it's also pretty much good enough, or else there'd be a lot more complaining about watching movies on computers or other 60hz devices.
A 30hz screen makes that harder, since you're going to have to repeat every fourth frame once, so the unevenness could be more noticeable since those frames are now twice as long instead of 50% longer. But I haven't tried one to tell you just how noticeable it would be.
Native 60fps content (sports is the most common one I know of) would require throwing away half the frames, but at least the remaining frames would all be shown for the same amount of time.
NTSC "video" content (stuff shot at 30 interlaced frames per second, so 60 fields per second, stuff like old TV shows) would work alright. You wouldn't be able to take advantage of deinterlacing methods that try to reconstruct 60 full frames, but it wouldn't be a huge loss.