The current taxonomy has family names like Canidae and Felidae. But Felidae has sub families like Felinae and Pantherinae. Roaring cats and non-roaring cats. My question is based on the fact that we can have common words for these families based in Latin: Felines, Canines, Pantherines. Can every family in the taxonomy be represented by this latter form? Is Pantherine a valid name? Is was created by dropping the a from the subfamily but caninine was created by dropping the da from the main family and replacing with in.
Is "Pantherine" a valid vernacular name for this group in English? Sure, although "panther" is much clearer ("Pantherine" in English would mean "like a panther", not "a panther"). Also, many family names are fairly obscure: even a pretty well-known group, like Apidae, would become "Apinine" in your scheme, which is less clear to me than "a family of bees".
Is it a valid family name? No. The International Code of Zoological Nomenclature (ICZN Code), which governs all scientific names for animals, says:
Article 29.2. Suffixes for family-group names. The suffix -OIDEA is used for a superfamily name, -IDAE for a family name, -INAE for a subfamily name, -INI for the name of a tribe, and -INA for the name of a subtribe. These suffixes must not be used at other family-group ranks. The suffixes of names for taxa at other ranks in the family-group are not regulated. [emphasis mine]
Family names that don't have an "-idae" suffix need to be corrected (ICZN Code Article 22.214.171.124).
Note that the ICZN Code doesn't cover other ranks in the family-group or above.