Well, I wouldn't disregard theory out of hand... I certainly have a good grasp of it. Granted, people call me smart, but I'm no Feynman either.
Regarding wire diameter, the principle is Ampere's law, in same class as Gauss's law and so on, which state that the sum flux through a boundary only depends on what's enclosed within that boundary. In the Amperian case, it's a closed loop curve, the enclosed area of which carries a current, and the flux is the magnetic field parallel to the path. Gauss's law applies to electric charge contained within a shell, the flux of which (electric charge) extends through, perpendicular to the shell; but also to gravity for another example, where the acceleration depends on the mass contained within a shell.
As with the Earth: if we could stand at the same altitude (distance from center), and imagine all its mass compacted into an infinitesimal point, we would experience the same acceleration as with all that mass distributed as it is currently. All that matters is that the mass is contained within a spherical shell beneath our feet. (The exact shape of the shell, does and doesn't matter, due to more rigorous conditions that I won't go into detail about; suffice it to say, this works best when the shell shares the same symmetry as the problem, i.e. a spherical shell for a spherical mass distribution, and the shell is larger than said distribution.)
So, too, the magnetic field around a wire, at a given distance from center, is independent of the wire diameter, so long as the wire radius is smaller.
And as long as the wire is cylindrically symmetric, the law still applies within it; the magnetic field drops smoothly from its value at the surface of the wire, down to zero at the center.
(The story is different for AC, where self-induction within the wire causes an opposing magnetic field, cancelling out the internal field: skin effect. Needless to say, this is more complicated, so I won't go into detail; just to note there's interesting things happening when wires or currents are changing over time.)
Which also gives us the tool to understand the other conditions.
Hollow: no field inside. In effect, all the surrounding currents cancel out here, but we would need a lengthy calculation to show it that way!
Other (non-circular) cross sections: same outside the highest peak, and same inside the lowest valley. The exact field now depends on angle as well as distance from center, so we can't say what the field is, at every point, so simply. We might need to use a numerical solution in this case (e.g. Biot-Savart law, replacing the integral with a Riemann sum).
Tim