x-rays
If I increase the accelerating potential in an X-ray tube the wavelength of the characteristic X-rays do not change.
Is this because when an electron beam strikes the target in an X-ray tube, part of its kinetic energy is converted into X-ray energy?
The angular distribution of the x-rays is basically uniform as you indicated, but there are beam restriction devices in use that aren't usually shown in a diagram, as in the typical diagram below.
Here is a more detailed drawing showing the isotropic radiation and a beam restricting device, which is less fancy that it sounds. Perhaps there are other fancier types.
The angled anode target are only angled around 10 degrees (6 to 15 degrees is what the source below tells me). The 45 degrees one often sees in diagrams must be an exaggeration. The angling helps with both heat dissipation and having a selected beam of x-rays be produced from a relatively large surface area on the target. The diagram below shows this better than I can explain it.
More information (and my source): http://en.wikibooks.org/wiki/Basic_Physics_of_Digital_Radiography/The_Source
Both the continuous spectrum and the characteristic lines are used; however, in some cases a filter such as aluminum can be used to remove low energy xrays that are not needed, so as to reduce xray exposure, as explained here: http://www.cyberphysics.co.uk/topics/medical/Xray.html
Best Answer
An X-ray tube creates a broad spectrum of X-rays due to Bremsstrahlung radiation and specific frequencies due to X-ray fluorescence. There's an example spectrum on the Wikipedia page about X-ray tubes. I assume you're talking about the fluorescence peaks since you refer to characteristic X-rays.
The fluorescence peaks are generated because the incident electrons collide with and eject electrons from the inner orbitals of the atoms in the target leaving the atoms ionised. When the ion recombines with an electron the energy of recombination is emitted as a photon. The energy of the emitted photon depends only on the energy of the atomic orbital that the electron is relaxing into, so it's a property of the atom. As long as the incident electrons have enough energy to ionise the atom the emitted photon will have the same energy.
So changing the energy of the incident photons may well change the intensity of the fluorescence peak, but it will not change the wavelength. Typically a commercial X-ray tube, e.g. for use in a diffractometer, will have the electron energy tuned to maximise the intensity of the required fluorescence peak (usually the $K^\alpha$.