Getting 15kHz video out of a system running Linux is straightforward, if perhaps non-obvious to someone not familiar with X's internals. It doesn't require any 3rd party software and is driver independent, which is handy.
It does, however, require that your hardware be capable of it. Not all hardware is. Unfortunately, in this case, your hardware probably doesn't. Most Intel IGPs of the "netbook" era won't do it. Even some nVidia graphics from that era wouldn't do it. Essentially all ATI/AMD will, though.
To answer the other questions, no, you're not likely to damage an LCD with a 15kHz signal. While the universe is great at inventing better idiots (designers, in this case) when somebody tries to make something idiot-proof, there's very little way to damage an LCD monitor in that manner. Also, no, you're not likely to find a mode that both your 15kHz SDTV can handle and your LCD monitor can handle unless you have one of the relatively uncommon (but by no means unheard of) PC LCD monitors that will handle 480i/240p 15kHz video.