Here is an answer to your question about monadicity, as it's too long for a comment. I will not fill in every detail, so if you follow along there will be several definitions that need to be expanded and diagrams that need to be shown to commute in order to verify everything, so let me know in a comment if you get seriously stuck.
In short, the first adjunction is not monadic, but the second one is.
For the first adjunction, the left adjoint is (or is isomorphic to, the way it's set up in the question) the contravariant power set functor $\newcommand{\powerset}{\mathcal{P}}\powerset : \mathbf{Set} \rightarrow \mathbf{Bool}^{\mathrm{op}}$, and the right adjoint $\Sigma : \mathbf{Bool}^{\mathrm{op}} \rightarrow \mathbf{Set}$ takes a Boolean algebra $A$ to the underlying set of its Stone space $\Sigma(A)$. Then the monad is $\newcommand{\U}{\mathcal{U}}\U = \Sigma \powerset$, which is known as the ultrafilter monad. Manes proved that the Eilenberg-Moore category $\newcommand{\EM}{\mathcal{EM}}\EM(\U)$ is equivalent to the category of compact Hausdorff spaces (by showing that it arises from a different adjunction between $\mathbf{Set}$ and the category of compact Hausdorff spaces, see e.g. VI.9 in Mac Lane's Categories for the Working Mathematician). From this, we can see that the comparison functor $\mathbf{Bool}^{\mathrm{op}} \rightarrow \EM(\U)$ is not essentially surjective, because it lands in Stone spaces, and $[0,1]$ is a compact Hausdorff space that is not homeomorphic to a Stone space. Therefore the $(\powerset,\Sigma)$ adjunction is not monadic.
The second adjunction is not stated correctly in the question, so I'll give the correct version of it. The functor $\newcommand{\Cstar}{\mathbf{C^*\mbox{-}alg}}\newcommand{\Top}{\mathbf{Top}}\newcommand{\CH}{\mathbf{CH}}\newcommand{\op}{^{\mathrm{op}}}\Sigma : \Cstar\op \rightarrow \Top$ takes a C$^*$-algebra $A$ to its Gel'fand spectrum $\Sigma(A)$, which is the set of maximal ideals equipped with the hull-kernel topology (the set of prime ideals is very awkward to handle even for $C([0,1])$ and doesn't work for a duality in this case). The functor $C_b : \Top \rightarrow \Cstar\op$ takes a space $X$ to the C$^*$-algebra of bounded complex-valued functions (this can be done with real-valued functions, as in Johnstone's book, but I prefer not to do it that way for compatibility with the noncommutative case. For a general topological space $X$, boundedness must be required, and the unbounded functions $C(X)$ do not form a C$^*$-algebra.) This time, the monad $\Sigma \circ C_b : \Top \rightarrow \Top$ is a form of the Stone-Čech compactification (this can be proved by verifying the universal property). So we write $\beta = \Sigma \circ C_b$ for short.
Now, we can prove that the $(C_b,\Sigma)$ adjunction is monadic as follows. If $(X,\alpha_X : \beta(X) \rightarrow X)$ is an Eilenberg-Moore algebra of $\beta$, then we have $\alpha_X \circ \eta_X = \mathrm{id}_X$, where $\eta_X$ is the unit. This implies that $\alpha_X$ is surjective, which, since $\beta(X)$ is compact, implies that $X$ is compact. It also implies that $\eta_X$ is injective, which, since $\beta(X)$ is Hausdorff, implies that $X$ is Hausdorff. Therefore $X$ is a compact Hausdorff space, $\eta_X$ is an isomorphism, and so $\alpha_X$ is required to be its inverse. This means $\EM(\beta) \simeq \CH$, and by Gel'fand duality the comparison functor $\Cstar\op \rightarrow \CH \simeq \EM(\beta)$ is an equivalence of categories, i.e. the $(C_b,\Sigma)$ adjunction is monadic.
