Good or bad, it does count as an installment in the franchise. So does Legendaryzilla, for that matter.
I believe it is referenced in a later movie.
Yeah. In Godzilla 2000 the monster from Tri-Stars 'Zilla is written off as another monster, mistaken for Godzilla. And in Final Wars the real Godzilla toasts the Tri-Star 'Zilla.