Is it possible to capture video signal waveform as audio for video replay? I am an electronic musician. For artistic purposes, I want to be able to record a video signal to a .wav file and replay that signal into a TV as video. It seems like it should be simple--if I reproduce the signal, it should turn out the same. Right?

I have an HDMI to RCA converter which works fine with my tv when used as expected in both NTSC and PAL mode. I take the yellow video cable from that and plug it into my audio interface with an RCA-1/4" adapter and record the signal as audio. When I replay the signal into my TV, it comes out like it's been subjected to Photoshop's motion blur horizontally at maximum, and in black and white. It's completely unrecognizable, looking only slightly better than static. If I use a recording from a black screen, I still get white in there. 

If I turn on monitor mode, bypassing, although apparently not completely, the audio interface and just route the signal through, it is still black and white, and while still totally unrecognizable, I notice localized changes if I move my mouse around.

If I just touch the two cables together it works fine.

So I'm wondering, is this a sample rate problem? If so, what sample rate would be necessary? Or is it something else?

Using Komplete Audio 6 interface

Sample Rates    44.1, 48, 88.2, 96 kHz
Bit Resolution  16, 24 Bit
The short answer to your question is "no".  The sample rate of video is typically measured in MHz or GHz.  If you have a 1080p24 signal at 8 bits per pixel, the actual frame is 2750 x 1125 pixels (according to digital test equipment manufacturer Kramer).  Multiplied by a 24 fps refresh rate, this results in 74,250,000 pixels / second.  Multiplied by 10 bits per clock tick per pixel, this is a bandwidth of 0.74 Gbps.  As a "sample rate", that is approximately 1000x higher than 16-bit at 48K.  Hence the answer "no".