Hi Shekinah,
Let's take a look at your analysis;
If an animal mates with another animal not of its exact species, the result will be a sterile creature (e.g. a horse mating with a donkey produces a sterile mule).
Well... that's almost true. Hybrid animals are usually, but not always, sterile. For reference, there is a list of some animal hybrids here;
Hybrid (biology) - Wikipedia
Note that many are fertile.
If animals of a given species mate and produce an abnormal offspring (i.e. a mutant), it also is sterile.
That's just not correct. A mutation is simply a novel DNA sequence. A "mutant" is an organism that has DNA sequences that did not originate with either parent. Generally the term "mutant" is only used to refer to those individuals whose mutation has caused them to differ noticeably from the rest of their population. Strictly speaking though, all of us are mutants, on average we have about two or three novel mutations, DNA sequences that were not present in our parents. These are the result of DNA copying errors that take place during the production of sex cells (meiosis).
Mutations do not usually prevent reproduction. Most mutations are neutral and have no effect. Since the rest of your argument follows on from this error, I would say that I have answered your questions; mutations do not usually make organisms sterile.
Mutate and Survive
"A curious aspect of the theory of evolution is that everybody thinks he understands it." - Jacques Monod