Get an input signal from LTI system output, find impulse response
Ok.  You are touching on the topic of deconvolution.  You can do this most easily by transforming the data into the frequency domain, doing some basic math and then converting back to the time domain.

For example for case 1, if you transform the signals you have
$ H(f) = Y(f)/X(f) $ so $h[n] = IDFT[Y(f)/X(f)]$

You can do similar manipulations to handle case 2.

I'm not sure case 3 is possible.  I think you need more information to produce x[n]

Here is a site that discusses deconvolution:  http://terpconnect.umd.edu/~toh/spectrum/Deconvolution.html 

This is a VERY simplified treatment of this question.  I'm not sure what level of detail is being requested in the posted question.