In my class notes we proved that the (Cech?; I'm not sure what it's usually called) complex $\underline{C}^\bullet((U_i), \mathcal{F})$ (Hartshorne Section III.4) is an exact cochain complex, where $(U_i)$ is an open cover of $X$. Hartshorne (in the first definition of Section III.4) defines the $p$th Cech cohomology group of $\mathcal{F}$ with respect to the covering $(U_i)$ to be $$ \check{H}^p((U_i), \mathcal{F}) = h^p(\underline{C}^\bullet((U_i), \mathcal{F})). $$ But wouldn't this lead to the Cech cohomology always being trivial? Having done examples in this area before, I've always ended up taking global sections of $\underline{C}^\bullet((U_i), \mathcal{F})$ and then taking the cohomology of the resulting complex. So from the examples I've done/what I've just said, I'd expect the Cech cohomology groups to be defined something like $h^p(\Gamma (X , \underline{C}^\bullet((U_i), \mathcal{F}))$, but they're not.
I think I'm missing something really obvious here. Thanks for any answers.
Edit: red_trumpet has answered my original question in the comments, though I'm still interested in Q1 and Q2 which I've written in the comments.
To expand on Q2, I now understand that in my class notes I have the sheafified Cech complex $\mathscr{C}^\bullet((U_i), \mathcal{F})$ which satisfies $$ \Gamma(X, \mathscr{C}^\bullet((U_i), \mathcal{F})) = C^\bullet((U_i), \mathcal{F}) $$ where $C^\bullet((U_i), \mathcal{F})$ is the global Cech complex (I've switched to Harthsorne's notation for this edit).
If $X$ is affine, then $\Gamma(X, -)$ is an exact functor. I presume in class that we proved $\mathscr{C}^\bullet((U_i), \mathcal{F})$ is exact, so by the identification above, $C^\bullet((U_i), \mathcal{F})$ too would be exact. Then putting extra conditions on $X$ such that Cech cohomology and normal sheaf cohomology agree, we get that $H^p(X, \mathcal{F})$ vanishes for $p>0$. Have we recovered a frequently stated "affine $\implies$ vanishing cohomology" result here?
I'll attempt to answer my own Q2. Let $X$ be a noetherian scheme with a finite open affine covering $(U_i)$ such that any finite intersection of the $U_i$ is affine $(*)$, and let $\mathcal{F} \in \mathsf{QCoh}(X)$. Then using the Leray spectral sequence and the fact that for an affine morphism of schemes $f: X \rightarrow Y$ and for $\mathcal{F} \in \mathsf{QCoh}(X)$ we have $R^pf_* \mathcal{F} = 0$ for all $p>0$, it can be shown that we have the canonical isomorphism $$\Gamma(X, \mathscr{C}^\bullet((U_i), \mathcal{F})) = C^\bullet((U_i), \mathcal{F}) \cong H^p(X, \mathcal{F}).$$ But then by what I've said at the end of my edit in the question, we indeed have for $X$ affine + noetherian + $\mathcal{F} \in \mathsf{QCoh}(X)$ + condition $(*)$ that $H^p(X, \mathcal{F})=0$ for all $p >0$, which is the result claimed at the beginning of section III.3, Hartshone p. 213, though actually Hartshorne does not state condition $(*)$; perhaps there's more work to be done if we drop it?