* partial answer

GeometricTransformation[g, RotationTransform[{u, v}, p]]

is sometimes equivalent to

Rotate[g, VectorAngle[u, v], p]

For example,

u = {1, 0};
v = {1, 1/5};
p = {1/3, 1/3};
g = Rectangle[];
Graphics[GeometricTransformation[g, RotationTransform[{u, v}, p]], Axes -> True]
Graphics[Rotate[g, VectorAngle[u, v], p], Axes -> True]

You could translate the figure being rotated back and forth.

GeometricTransformation[g, RotationTransform[{u, v}, p]]

is equivalent to

Translate[Rotate[Translate[g, -p], {u, v}], p]

For example,

u = {1, 0};
v = {1, 1/5};
p = {1/3, 1/3};
g = Rectangle[];
Graphics[GeometricTransformation[g, RotationTransform[{u, v}, p]], Axes -> True]
Graphics[Translate[Rotate[Translate[g, -p], {u, v}], p], Axes -> True]

This works in both 2D and 3D.