Keypoint position and scale in SIFT
The $\sigma$ parameter is both. The Gaussian function can generate a scale-space where $\sigma$ is the scale parameter. It doesn't mean the image is scaled, instead it is the scale at which the features are being evaluated.

For example, with higher $\sigma$ the image is more blurred and therefore only larger image features contribute to the gradient histogram. SIFT is made scale-invariant by choosing the greatest feature response from a set of responses from different scales. The Gaussian function is used because with increasing scale one can guarantee no new local extrema (features) are introduced.

I'm not sure about the interpolated extrema, but I guess it is to obtain sub-pixel accuracy.