You do realize that if you double the frames per second you're using twice the amount of memory.
No it doesn't. Doubling fps has no impact whatsoever on memory use. It doesn't even increase the amount of needed bandwidth because you do not need additional textures just to show images faster.
Doubling the number of pixels, pretty much doubles the amount of memory used.
No it doesn't do that either. Framebuffer, Stencilbuffer, z-buffer etc. will be twice as big but they only consist of a small part of the GPU memory. Textures are the biggest memory users and they have per se no connection to render resolution. (You may want more detailed textures in higher resolutions but this is a different matter).
Memory doesn't have a big correlation with render resolution and none with fps. The biggest drawback for both of them are the amount of pixels the GPU has to compute.