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<blockquote>What is a typical value for capacitance of a real inductor?</blockquote>

"Typical" capacitance is a useless to design circuits with.  It will also vary considerably by inductor size and geometry.

The only real answer is in the datasheet of whatever inductor you are considering using.  There is no substitute for reading the datasheet.  If the datasheet doesn't tell you the equivalent parallel capacitance, or the self-resonance frequency, then you can't use that inductor in serious applications.  Take a look at datasheets from reputable inductor manufacturers, and you will see such specifications.

<blockquote>Does this internal capacitance exist only under AC and not DC?</blockquote>

Stop and actually <i>think</i> about it.  The capacitance is a function of the geometry and materials.  A certain amount of charge is stored per volt, which is the definition of capacitance.  When you buy capacitors, do their values change depending on whether you use them with AC or DC voltage applied?

Real capacitors aren't ideal, so they only act like capacitors up to some frequency range.  Still, that doesn't seem to be what you are asking about.  Some ceramic capacitors exhibit reduced capacitance (less charge stored for the same voltage increment) at high voltages, but that also doesn't seem to be what you are asking about.

A 100 pF capacitor is 100 pF, whether it's sitting in a box, has DC voltage applied to it, or 10 Hz AC applied to it, assuming maximum voltage specs aren't exceeded.  The parasitic capacitance of an inductor is going to be quite constant, probably to past the self-resonance frequency of the whole inductor.

It seems you are confusing a capacitance with its impedance.  The impedance magnitude of a capacitor does depend on frequency, being inversely proportional to that frequency.