Hello lanak, welcome to the forums.
1. Although absorption energy of vibrating bonds are what give rise to spectral information, a better way to frame it would be to say that vibrational modes are responsible, not specific bonds. I.e., an absorption event usually involves more than one bond at a time. Consider the symmetric stretch of water. This involves the coordinated stretching of both O-H bonds simultaneously, not a single bond.
It's a little hard to understand what you're asking in the remainder of your question, but you might want to see the wikipedia article on Fourier Transform Spectrsocopy. Some spectroscopic techniques measure the spectral event at each wavelength independently and then build a spectrum from that information. Fourier Transform essentially allows you to measure all spectral events simultaneously and then transform all that information into a single spectrum via a mathematical technique called Fourier Transform. NMR and FTIR are two of the most common spectroscopic methods that work this way. The advantage is that instead of measuring each probed wavelength individually (which takes a whole lot of time), you can do it all at once.
https://en.wikipedia.org/wiki/Fourier_transform_spectroscopy2. Refer to above. Each bond can be involved in more than one transition. In water, each OH bond can participate in symmetric and asymmetric stretches, as well as bending motions. If molecular vibrations were governed by classical physics, then supplying light with a large amount of energy would simultaneously excite every possible vibration. But molecular vibrations are governed by quantum physics, and there is a resonance condition - there is no absorption unless the incident light has exactly (to an approximation) the same energy that is required to excite the vibration.
3. The wavenumber in spectroscopy is one of those rare instances of a non-SI unit persisting. Its popularity is primarily historical. It's really only used in spectroscopy, most notably FTIR and Raman, but also in some instances in electronic and other spectroscopies.
https://en.wikipedia.org/wiki/WavenumberFormally it is an expression of the frequency per fixed distance. The wavenumber actually isn't the unit - but it is often used this way in common parlance. The most frequent expression of the wavenumber is the cgs inverse centimeter, which I learned recently is also called the Kayser.
4. There are many things that contribute to bandwidth in FTIR. The most commonly encountered is inhomogeneous broadening, arising due to slightly different chemical environments around a vibrating molecular mode. A recognizable example is the OH stretch of water or other O-H containing molecules, which is typically very broad. The broadness derives from the fact that there are lots of slightly different O-H environments due to all the hydrogen bonding, and the spectrum represents an average of all of these. Note that if you measure these in the gas phase, or in dilute (non-hydrogen-bonding) solution, the O-H stretch narrows up considerably, reflecting a more homogeneous environment for O-H stretches.
5. Electronegativity doesn't directly correlate to band intensity. The intensity is (mostly) due to the degree of dipole moment change during the course of the vibration. Bonds that involve atoms of high electronativity differences will tend to have stronger peaks because the dipole moment change is larger for these bonds during the course of a stretch. But bear in mind there are other things that affect both the frequency of the transition and also the intensity and broadness of the resulting spectral absorption peak.
A good introductory treatise of some things that affect band position, intensity, and broadness can be found here:
http://chemwiki.ucdavis.edu/Core/Physical_Chemistry/Spectroscopy/Vibrational_Spectroscopy/Infrared_Spectroscopy/Infrared%3A_Interpretation