This is perhaps a question that can't be answered, so if that's the case, that's totally fine, I just really like linguistics and stuff like that. Is Storm- eligible for -fall since -fall is for elemental and tree based prefixes? If so, what would be the cultural difference between a cat named Stormfall and a cat named Stormstorm?
Storm- is indeed eligible for -fall! As to the Stormfall/Stormstorm difference (similar to Cloudfall/Cloudcloud), the -storm and -cloud suffixes have abstracted from their literal meanings of "turbulent like a storm" and "serene like a cloud". -fall (and its siblings) mean "this cat is very much like their namesake in demeanor".
Imagine a pair of sisters, Stormpaw and Cloudpaw. Cloudpaw is chill, befitting -cloud, but there's nothing much to her that also screams "cloud". She's not terribly fluffy, she doesn't drift when idle, etc, etc, so she'll probably become Cloudcloud as a warrior. Stormpaw, on the other hand, is huge, looming over other cats and filling the atmosphere with tension. She has a voice like Horoa's and a temper to match. In movement she stirs up dread, crashes down in a fury, and leaves behind a trail of destruction. Sure, the leader could name her Stormstorm and that'd be fitting enough with her mood, but everyone knows this is a cat deserving of the name Stormfall.