Already mentioned, but this isn't really a sign of a "weak female character" at all. If she didn't have some kind of trauma after what happened such a short while ago, she'd be one of those exaggerated badasses whom you can't even sympathize with. She'd basically be Conan the Barbarian or something, at the age of 7. Very realistic.
...plus Kain, since part of the reason Rosa gets kidnapped is related to Kains feelings for Rosa as well. So yeah... if you basically changed the whole story around so much that Cecil and Rosa didn't have any relationship from the start, and both Kain and Cecil fell in love with Rydia... then things would obviously have been different, since you've basically changed some of the greater plot-points around completely and reveresed the roles of two characters. What would have happened in such a fanfic does not change what happens in the actual game though, and the "damsel in distress" of FFIV is no doubt Rosa, not Rydia.
As you luckily mentioned, a 7 year old girl getting knocked out by an earthquake plus using summoning powers beyond her limit is somewhat understandable. Somewhat.
I guess the same argument can be used for the guards too. If Rydia had been adult by that time, those guards would have been charged to crisps. But a 7 year old girl against a bunch of trained soldiers? That might be asking for a tiny bit too much again.
It's not Rosa's fault she gets the desert fever and gets kidnapped etc, but I think her cases are still more "damsel in distress-like" than the cases with Rydia. It still gives her a role in the story that I feel is less interesting than Rydia's. That's my take on it.
I want to get in her panties!