When shooting with a rectilinear ultra wide, close subjects seem distorted and enlarged, but far subjects seem pushed away and small. Does that mean there is a sweet spot in between where proportions appear "correct"? If so, is there a formula to calculate that spot?
Is there a sweet spot in distance to a subject when shooting with an ultra wide angle lens?
It has very little to do with focal length directly.
Perspective distortion is caused by one thing and one thing only: Shooting distance.
The reason we notice it more when using very wide angle lenses is because we tend to place the camera closer to the subject than when we use longer focal lengths with narrower angles of view.
If you wish to frame a certain subject using a certain perspective, you should find the appropriate distance for that perspective and then use a lens with the focal length needed to frame that subject the way you wish at that distance.
Enlarging the relative differences between things that are close and things that are far is just one of the results of using the wider angle of view provided by short focal lengths, just as compressing the relative differences in distance between two objects which are both some distance from the camera is the result of using lenses with narrower angles of view.
On the other hand, the distortion that is the result of a rectilinear wide angle lens will apply equally to near and far objects. Things in the corners seem to get stretched towards the corners in order to make straight lines in the field of view appear straight. This is just as true of mountains or large trees in the distance as it is of a much smaller object much closer to the camera when both are in the same corner of the angle of view. We just seem to notice it more when it is a person's head than when it is a cloud in the sky.