Questions about Evolution

[Anomalocaris]

I was truly fascinated by the article The RocketSurgeon posted about how scientists can determine if a fossil belong to a biped or quadruped by determining the position of the spinal cord in the skull.

This raised a few questions that you guys might like to explore.  
1. When examine two or more fossils, how do scientists determine whether they are looking at two separate species within a genus or two variations of a single species?  
2. Can a mutation result in offspring of a different species from the parent?

1.  There would be no way to determine that with any degree of certainty unless two specimens contain overlapping parts, although it is possible to make informed guesses.   Let's say a future peleontologist found two sets of fossils.  Both belong to Homo sapiens, but one set only has bones from the waist up, the other from pelvis down.   Based on the two sets of fossils alone, it would be impossible to determine if they belong to the same species.  But the paleontologist can do some more digging to make an informed guess.   For example, she might look through academic papers and discover one of her colleagues had found a complete set of bones from a monkey.  Our paleontologist can compare the two sets of fossils she found to the complete monkey skeleton, and discover Both sets bear very strong resemblance to the upper and lower bodies of the monkey.   So she might conclude the sets of fossils she has found each belongs to creatures bearing strong similarity to monkeys.   She may further note that the two sets of fossils have traits which suggest they share common traits different from those of the monkeys.   She may for example note that based on the skull of one set of fossils, and the pelvis of the other set, that both sets belonged to creatures that frequently walked upright.   From this she may conclude the two sets of fossils may be related to each other more closely than either do to monkeys.  Monkeys don't habitually walk upright, and both monkey skulls and monkey pelvis shows this. She might further notice the spine of one set of fossils is approximately the right size to fit the pelvis of the other.  From this she might conclude the two sets of fossils belong to animals of broadly similar size.  Finally, she might notice the two sets of fossils came from roughly the same geological era and were discovered relatively close to each other.  So from here, she might guess they belonged to perhaps the same population.  Nothing for sure, but certain an educated, rather than wild, guess.

If they have significant overlapping parts, then it is possible to compare the details of same bones in two different specimens, and arrive at a conclusion that the two animals they came from were closely related or not.   If there are enough fundamental similarities then they were likely members of the same species.    The degree and depth of similarity is not subjective.  There are technical ways to analyze similarity to determine if the simialrities arose because the two specimens were closely related, in which they might be the same species, or if the similarity arose separately because two unrelated animals shared some similarities in life style, in which case they are probably not the same species.

2.  Technically yes.  It is possible.   But it is extremely unlikely.  Typically the formation of a new species is a slow gradual process that occur over many generations.  There is typically no generation where speciation occur.   But when compared across many generations it is clear the distant descendant is no longer the same species as the distant ancestor.     Speciation could theoretically occur over a single generation as a result of hybridization.  In this case, two different species interbreed, and the result could not freely interbreed and produce fertile offsprings with either of the parent species, but could interbreed freely amongst themselves.  In this case speciation occurs over a single generation.   But it is extremely unlikely.