Well, interestingly enough each of the links in your inquiry is to the Diccionario de la lengua española definition, so I am guessing that the insight each provides is not sufficient. Here's my take on each of the words you are studying:
muerte, fallecimiento, deceso
These three words are nouns that can be used in the singular as well as the plural. They are largely interchangeable as in
provocó la muerte / el fallecimiento / deceso (caused the death)
noticia de la muerte / del fallecimiento / deceso (notice of death)
fecha de muerte / fallecimiento / deceso (date of death)
prematura muerte / prematuro fallecimiento / deceso (premature death)
reciente muerte / fallecimiento / deceso (recent death)
condolencias por la muerte / el fallecimiento / deceso (condolences on ...)
and can be used as an idea or an event. (See my comment in response to Pablo Chávex's answer.)
Nevertheless, it is likely certain turns of phrase require one or the other. I am not a native speaker of Spanish, but it appears that "muerte" must be used in the following phrases (or is far more common):
pena de muerte (death penalty)
condenado a muerte (condemned to death)
amenazas de muerte (death threats)
sentencia de muerte (death sentence)
I did not find any fixed/set phrase or collocation for "fallecimiento" that could not also be used with "muerte" or "deceso," but I am not a native speaker so if anyone knows otherwise, please correct me.
On the other hand, I did find a few phrases with deceso that did not appear to be interchangeable with "muerte" or "fallecimiento" (or if they were, the meaning was slightly more literal and less idiomatic than the phrase you see below with "trágico").
seguro de decesos (funeral insurance)
trágico deceso (tragic passing)
As mentioned, "deceso" is interchangeable with "muerte" and "fallecimiento," but it is far less frequently used in general.
If you look up "death" on Wikipedia and then link over to the Spanish page for the same, you will be taken to Spanish Wikipedia's page on "Muerte." (And if you search for "Fallecimiento," you'll be redirected to the page on "Muerte.")
All three nouns can be traced back to a Latin root.
The famous play Death of a Salesman is translated into Spanish as Muerte de un viajante.
Last, but not least, we have "fallecer," which is a verb and the one from which "fallecimiento" derives. It is synonymous with "morir." Both mean "to die." However, according to Fundéu (Fundación del Español Urgente), there are some who believe that "fallecer" is to be used only for "natural" death. According to the Royal Spanish Academy, this is not true; "fallecer" can be used for any type of death. That said, the Fundéu page also adds that "fallecer" should be used with people and not animals. From this, I assume that "morir" is the more appropriate term to use for animal deaths. Again, I am not a native speaker, so if any of you see a need to correct what I've written, please do so.
Collocation information was retrieved from Linguatools. For more collocations (I only looked at the first page for this answer), visit the following links:
By the way, "muerte" had 2,756 collocations; "fallecimiento," 306; and "deceso," 22.
The nouns "muerte," "fallecimiento," and "deceso" are largely interchangeable, but "fallecimiento" seems to be more formal and there are set phrases with "muerte" that would sound odd if you replaced them with "fallecimiento" or "deceso."
With the exception of "fallecer" all three are nouns and all three can be found in the singular as well as the plural.
According to some schools of thought, "fallecer" is only to be used with people and not for animals. The verb "morir" can be used with both.