I happen to be a Horn Player and can play both Natural and Valved Horn. It isn't exactly simple, as you clearly have found out. Unless you find a period orchestra, it is generally not going to be played on natural horn. In Paul Dukas Villanelle, he calls for natural horn in the beggining section, though I have only found one (amatuer) horn player actually utilize a natural horn for the opening.
That being said, the harmonic series ends up manifesting itself differently for every crook, and that some keys favor high notes and others favor high notes. There were conventional crooks in a low Bb, a low C, D, Eb, E, F, G, Ab, higher Bb, and higher C. The other keys are fairly rare but could be obtained by adding a piece of lengthening. A and B are supposedly reachable in both octaves.
For simplicity I'll speak in notated pitch, not concert pitch since the crooks greatly vary the potential pitch range.
The open notes with best quality in the treble clef are usually (in rising order starting around C two ledger lines below treble) C, E, G, Bb, C, D, E, G, G#, Bb, B, C and in some lower keyed horns, E and D above that (there are several more than can be hit but will ultimately sound out of tune unless under specific circumstances like Gb above the stave which is extremely flat). I would suggest using notes notated above the treble clef sparingly due to endurance and pitch issues, this becomes increasingly true as the pitch of the crook rises. In keys higher than F, The G above the stave is the highest note, except for high C, whose highest note is E. G and F below the treble Stave are also possible and used with frequency. They are rarely notated in Bass clef.
In the bass clef, the notes become farther apart and harder to hit with accuracy and clean articulation. It usually used just for sustained chords in the lower horn parts if they go really low. There is a gap in the range that can't be reached by stopped hand technique between treble and bass clef, so the highest note is C (notated three ledger lines below bass clef. Yes this is a strange convention of the era) and the G below that, and it is possible to move chromatically down descending but this is difficult in Horns other than E, Eb, and F. Lower notes are possible but typically aren't practical.
A good horn player can play chromatically from C two ledger lines belowthe treble, or maybe even the G below, if they are excellent at stopped hand technique, but the sound quality will be quite bad on quite a few notes. In ascending order B below 2 ledger C, Eb, F# to a lesser extent, A, B, C#, Eb, F, F# to a lesser extent, Ab, and A to a lesser extent are the good stopped tones that sound nice.
Wagner mostly wrote parts that were possible on Natural Horn, but are almost always played Valved Horn or Wagner Tuba.