The place where you changed the frame rate is for the entire flash movie.
You cannot defire framerate for an individual symbol.
On the suggestion node, if its possible, you can remove some of the frames in the gif animation symbol imported to timeline, which may increase the frame rate in an alternative way without changing the frame rate of the whole flash.
But be careful not to remove too many frames otherwise, there will be jerky animations.
Ok, but what if I wanted to slow the .gif animation down instead of speeding it up? Would there be a way to do that? (one .gif file seems to be playing much faster in the Flash movie than when I open it alone).
Yes, do the reverse trick.
In the gif imported to flash, make each screen of the gif in 2 frames instead of 1.
So selecte each keyframe in timeline and press F5 to insert a frame.