I think that what you’re missing is that an open cover of a compact set can cover more than just that set. Let $X$ be a topological space, and let $K$ be a compact subset of $X$. A family $\mathscr{U}$ of open subsets of $X$ is an open cover of $K$ if $K\subseteq\bigcup\mathscr{U}$; it’s not required that $K=\bigcup\mathscr{U}$. You’re right that $\bigcup\mathscr{U}$, being a union of open sets, must be open in $X$, but it needn’t be equal to $K$.
For example, suppose that $X=\Bbb R$ and $K=[0,3]$; the family $\{(-1,2),(1,4)\}$ is an open cover of $[0,3]$: it’s a family of open sets, and $[0,3]\subseteq(-1,2)\cup(1,4)=(-1,4)$. And yes, $(-1,4)$ is certainly open in $\Bbb R$, but $[0,3]$ is not.
Note, by the way, that it’s not actually true that a compact subset of an arbitrary topological space is closed. For example, let $\tau$ be the cofinite topology on $\Bbb Z$: the open sets are $\varnothing$ and the sets whose complements in $\Bbb Z$ are finite. It’s a straightforward exercise to show that every subset of $\Bbb Z$ is compact in this topology, but the only closed sets are the finite ones and $\Bbb Z$ itself. Thus, for example, $\Bbb Z^+$ is a compact subset that isn’t closed.
It is true, however, that compact sets in Hausdorff spaces are closed, though a bit of work is required to establish the result.
I don't know how useful this will be, but I have some lecture notes that motivate the last three things on your list by first reinterpreting the finite dimensional spectral theorem in terms of the functional calculus. (There is also a section on the spectral theorem for compact operators, but this is just pulled from Zimmer's Essential Results of Functional Analysis.) I gave these lectures at the end of an undergraduate course on functional analysis, though, so they assume familiarity with Banach and Hilbert spaces.
Best Answer
You also need to be careful, as there are two possible and used definitions of locally compact: (1) every point has a compact neighbourhood; (2) every point has a base of compact neighbourhoods. The advantages of the second definition are: (i) it is in the spirit of definitions of local properties, and (ii) it is very often this property which is used in practice. The two definitions agree for Hausdorff spaces.
Of course the previous comments are on the ball.