I'm sure AWK will have a bunch of links to more complete explanations, but the basic idea is that you subject a molecule to a pulse of magnetic energy which excites the nuclei to a higher energy spin state.  As the nuclei relax to the lower energy spin state, the spectrometer records the resultant energy emissions.  These specific energy of these emissions depends on the electronic environment of the nucleus that was excited, that is why you see a difference in the NMR spectrum for electronically different nuclei.

I just checked in a textbook and it says that the emissions are what are detected.

Absorbtions are detected in other forms of spectroscopy like IR though.

I'm sure AWK will have a bunch of links to more complete explanations, but the basic idea is that you subject a molecule to a pulse of magnetic energy which excites the nuclei to a higher energy spin state.  As the nuclei relax to the lower energy spin state, the spectrometer records the resultant energy emissions.  These specific energy of these emissions depends on the electronic environment of the nucleus that was excited, that is why you see a difference in the NMR spectrum for electronically different nuclei.

Actually, you subject the molecule to a pulse of radio energy while applying a strong magnetic field.  The magnetic field will create a difference in potential between nuclei whose spin is aligned with the magnetic field and nuclei whose spin is aligned against the magnetic field.  The radio wave pulse will excite all of the molecules to the higher energy state (spin against the applied magnetic field).  Then the NMR machine detects the radio waves emitted when the nuclei return to the ground state.  (Radio waves are used because they correspond to the energy difference between nuclear spin states.)

Different nuclei will emit radio waves of different frequencies because the size of the energy difference between spin states depend on the strenght of the magnetic field felt by the nuclei.  Although the applied magnetic field is uniform throughout the sample, the magnetic field felt by the individual nuclei is affected by shielding from electrons surrounding it (as well as magnetic fields induced by neighboring electrons).

This question I'm not so sure about.  Looking over my notes on NMR, I can offer this as a guess:

NMR spectrometers use a broad pulse of radio energy to excite all magnetically active nuclei simultaneously, then record the emissions on an intensity-v-time graph.  The computers in the spectrometer then subject this data to a Fourier transform which converts the intensity-v-time graph to an intensity-v-frequency graph based on the decay of the signals.  This is probably easier than producing or recording radio waves of a specific frequency because radio waves have such long wavelenghts and would require uneccessarily large equipment to produce and/or record radio waves of a specific frequency with the neccessary degree of accuracy.  In addition, using the Fourier transform method probably makes the spectrometers cheaper to produce since they require less sophisticated radio wave generators and detectors.

However, protons are not only spinning, protons are also precessing at a frequency, called Larmor frequency.

You can find that the NMR spectrometers have their field strengths (in Tesla).
e.g. 7 Tesla will give 300 MHz of precession frequency of protons
Larmor frequency = gyromagnetic ratio (in MHz/Tesla) x Bo (in Tesla)

As the no. of protons in ground state is higher (assume the external Bo is pointing upward, z-axis), the net magnetization is pointing upward also.
Then, applying a 90 degree pulse (for example) using radio frequency.
(This pulse can generate a continous radio frequency in particular range, but I am not familiar with this principle.)

This pulse will exite the gd. state protons to ex. state. Not all protons are being exited for 90 degree pulse, it will only make the population between 2 states are the same.
The net magnetization is now on xy plane, and the receiver (on xy plane) convert the signals to peaks by FT.
(I am not quite familiar with this part, just briefly describe it)