Magic eraser is a mild abrasive. If the stains are embedded in the clear coat, you'll have to remove the clear coat to get the stain off. If the stain is in the paint, then you'll end up going to the primer. I imagine at the very least it will degloss the clear coat and outer layer of the paint, so if you don't want this then I'd test it somewhere inconspicuous.
Scratch-X is good for fine scratches, and didn't cause too much deglossing of the paint I used it on. I followed up with a good paste wax.
I think repainting this frame, from what I see, is overkill. Plus, as mentioned, a good paint job is going to be greater than the value of the frame/fork. the prices I see for a good prep/wet paint/decals start at $500 and go up from there.