To answer your question about paraphyletic groups:
Clades are a synonym for "monophyletic" groups, which are to be contrasted with paraphyletic groups. Below is a good diagram explaining the difference:
There are two very common examples of paraphyletic groups: "fish" (in blue) and "reptiles" (in green). Diagram below:
Evolutionary biologists prefer grouping organisms into monophyletic groups (clades) because this type of grouping is more informative and useful. Monophyletic groups are defined by the species in the clade all sharing a set of characters, or features ("synapomorphies"), that are different from that of the ancestral species. One fantastic example of a monophyletic group is mammals; some unique shared features include having hair, mammary glands, and three middle ear bones. This kind of grouping is informative because by looking at what kind of features a species has, you can place it into a clade AND know which species are most closely related to it.
Unfortunately, a lot of Linnean classification isn't based on cladistics (grouping species into monophyletic groups), so you end up with a lot of paraphyletic groups, such as fish, reptiles, and dinosaurs. A better way to define the groups in this diagram would be to call all those species in the clade "Gnathostomata" (jawed vertebrates) and then the clade for green+birds "Sauropsida." So why do we even use Linnean classification if it's not helpful in clarifying evolutionary relationships? It's just because it's been stuck with us for a while now... eliminating the concept of "fish" and "reptile" would be pretty difficult.
I think the Fossils list tried to balance Linnean classification and cladistics in its organization of the genera. Invertebrate classification (in real life) are just a cladistics mess, but the vertebrates on the list are generally in monophyletic groups.