Felis Pinnula or Felis Avia
The one for giant spiders is close. Technically, you should probably choose one of the many genus names. But "Arachnid" paired with anything denoting large size- Gigans, Mega, Titan, etc.- would work. You could or even should even use those variants to describe the various distinct ones.
OTOH, you could create a new genus- saaaaay, "Titanarachnea"- to denote the genus of ALL the giant sized spiders of fantasy and then you could have fun with the species names to distinguish them. In this case, the common giant spider would be Titanarachnea vulgaris.
Something like a phase spider would be Titanarachnea aetherius. The biggest species in your campaign setting could be Titanarachnea Rex.