I am ignorant about laser physics, but i have this idea of collimating monochromatic light with an optical fiber/cable that is cone shaped: at inlet the cone has a wide diameter (several hundred microns - like an ordinary optical fiber, which might be used as a light source), on the other end it narrows to almost light wavelength diameter (e.g. 1.1 wavelength) and next for some distance it keeps this small diameter constant (it's not a cone at the end any more). I imagine that light would be squeezed in such a device to be almost unidirectional, with good beam quality. It would also greatly increase optical power density at its end, which would probably make it difficult to operate continuously. Maybe using hollow, cryogenically cooled, small cone angle and long fiber would help. My question is: practical challenges aside, are there some fundamental physics laws that make such a device pointless? Will be the outgoing light squeezed into narrow, good quality and much more intense beam? I have a feeling that my intuition is naive in some way, but I don't know how.
No this won't work.
are there some fundamental physics laws that make such a device pointless?
Yes. You're not considering diffraction.
Because of diffraction, the light exiting a fiber will actually diverge at a larger angle, the smaller you make the output aperture.
So in fact, to make a fiber taper that improves the collimation of the output beam, you need to taper the fiber from smaller to larger, not larger to smaller.
For a device that works similar to a (multimode) optical fiber that can be used for collimating a laser beam, you should investigate GRIN lenses.