9
This is a rather unusual and interesting case of aromaticity, which has been given a special name: homoaromaticity.
The Wikipedia page does a quite nice job of explaining what's going on. As you state, protonation of cyclooctatetraene generates a $\ce{C8H9^{+}}$ cation containing an $\mathrm{sp^3}$-hybridised carbon atom between all the other $\mathrm{sp^2}$-...
3
Setting aside the arguments over umlauts, yes you can add the nitrogen atom to the Hückel matrix; in fact, theoretically you should do so and allow the matrix eigenfunctions and eigenvalues to decide whether the nitrogen atom in this case is really conjugated. The carbonyl oxygen can also be coupled in.
But there is a catch. Since the nitrogen or
oxygen atom ...
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