I am 98% sure your problem is due to "Symbols". For some reason Macromedia Flash 8 Professional and lower versions do not export symbols properly if you choose gif as the export type (there might be a way but if there is, I don't know about it). Just put the animation frames in the root to fix your problem. It may make it less convenient in terms of making the animation but, it does not in any way lower the quality of animation. This also means that you cannot use script to animate your objects.
You are using a lower quality of exportable gif type (this is obvious due to the clearly visible colour bands of separation for your gradients). If you want better quality then change "Dither" to "Diffusion" and make sure gradients are fully enabled (there is a tickbox "Remove gradients", make sure it's unticked) in the export settings for gif (see attached image).
What version of Macromedia Flash do you have? I have Macromedia Flash 8 Professional and that version has those options in "publish settings" which you say you could not find but I'm sure even MX had it (see attached image).
You should fix your image bounds or document dimensions, there a quite visible white bars on the right and bottom.
I'm not 100% sure what those other options in the right screen shot do so if you play around with them you may be able to get better quality than with just unticking "Remove gradients" and setting "Dither" to "Diffusion".