I can’t claim to be completely up-to-date on felid classification, and since most of the on-line info doesn’t give much in the way of references it’s hard to be sure which are most authoritative.
Here’s one classification:
Subfamily Felinae
Subfamily Patherinae
This one is unusual in including the Cheetah in the Felinae instead of in its own subfamily.
Here’s another that continues to recognize the Cheetah as a separate subfamily.
With respect to the OP, “cats that roar” are in fact a natural grouping, that is, the genus Panthera, including the lion, tiger, leopard, and jaguar. However, “cats that purr” (or, better, “non-roaring cats”) is not a natural grouping. Some non-roaring cats are more closely related to roaring cats than they are to other non-roaring cats; therefore one cannot say this criterion is used in classifying cats.
As for the laying/crouching distinction, I doubt whether this has any validity at all. If there is any difference in tendency to do this, it is likely based on the size of the animal (a crouching stance perhaps being more difficult to maintain by a larger animal) than by actual relatedness.
I don’t think she’s going to be getting this published in any of the more reputable taxonomic journals anytime soon. 
There are no objective criteria for the recognition of genera; basically a genus is whatever a taxonomist says it is. Because the small cats are pretty homogeneous morphologically, and were not easy to separate clearly on that basis, the tendency was previously to put them in the same genus. However, now that genetic data is available, it has become possible to identifiy different lineages within the group, so that a genus can be recognized for each lineage.
Genetic data however can cut both ways. While in many cases it can lead to splitting of genera, in others it can lead to lumping of previously separate genera when they are found to be more closely related than previously supposed.
As for the second question, I don’t know what the consensus is in detail; but the tendency these days is certainly to split up Felis into several genera.