Magnetic field of a 25 micron thin wound up copper ribbon?
The magnetic properties of copper are much the same as air, or vacuum. The relative permeability is 0.999994, so not unity, but pretty close. Given that transformer ferrites and iron can have relative permeabilities of 1000 and 10000 respectively, we can call a 6ppm difference from unity one!

Although you can look up 'spiral inductor' and arrive at a calculator page like this one, I doubt that the simple models they use would cope well with the extreme thickness/width ratio of conductor you propose, though they may get you into the right ballpark. You would have to do your own calculus, starting from Biot Savart. Accuracy at higher frequencies would still be suspect due to the current crowding that would occur.

The thermal properties, given that the copper extends to both edges of the coil, would be better than for a conventional wire-wound copper coil of the same wire cross section and coil dimensions, as heat could be lost directly to the outside edges without having to make its way across insulation for the inner turns.

The most significant difference in effective magnetic properties compared to a wire-wound air-cored coil would be the fact that changing magnetic fields will induce circulating currents to flow in the width of the tape, leading to high eddy current losses. This effect is the source of the current crowding I mentioned when discussing inductance calculations.