As far as I can tell, every example except the last is representation theory, understood in a suitably general sense. For example studying the eigenvalues of an operator is the same as studying the corresponding representation of $\mathbb{C}[X]$ or some suitable enlargement of it (e.g. the $C^{\ast}$-algebra generated by $X$), so 1, 2, 3, 8, and 9 are morally special cases of 4. 4 and 5 are morally special cases of the representation theory of various types of rings and algebras, while 6 and 7 also fall under this category thanks to the existence of various types of group algebras.

(I can't tell if you know this or not, but the spectrum of a commutative ring is also just representation theory. After all, every prime ideal $P$ of a ring $R$ gives rise to a homomorphism $R \to R/P \to \text{Frac}(R/P)$ which may be understood as a one-dimensional representation of $R$.)

As far as I know, spectral sequences and spectra in homotopy theory are unrelated to spectra in the above sense.