Are these strengthenings of Serre-Swan and Gelfand-Naimark true?

Take $X=\omega_1$, in the order topology. Any real or complex valued function on it is bounded so $C(X)$ and $C(\beta X)$ (here $\beta X \simeq \omega_1 +1$, of course) are isomorphic as rings (and as (real) algebras as well, an isomorphism of rings $C(X)$ is always an $\Bbb R$-algebra isomorphism too; I haven't studied the complex case well enough, but the real case is classical.) So in general locally compact Hausdorff spaces you cannot distinguish a space from its (C-S) compactification on ring or algebra structure alone...

Conjecture 1 is true.

The main technical content of the proof is that over a compact Hausdorff space, every real or complex vector bundle is a direct summand of a trivial real or complex vector bundle; see this math.SE question (which doesn't have a proof but does have a comment with a link to a proof).

When $X$ is compact Hausdorff, the C*-algebra structure of $C(X, \mathbb{C})$ can be recovered from its $\mathbb{C}$-algebra structure, so the $\mathbb{C}$-algebra structure determines $X$.

To see this, first observe that the range of a function $f : X \to \mathbb{C}$ can be recovered by considering its spectrum $\sigma(f)$ (the set of $\lambda \in \mathbb{C}$ such that $f - \lambda$ fails to be invertible), which only depends on the $\mathbb{C}$-algebra structure. From the spectrum we can recover the norm as the spectral radius

$$\| f \| = \sup_{\lambda \in \sigma(f)} \| \lambda \|$$

so the norm can be recovered from the $\mathbb{C}$-algebra structure. Next, we can also get the *-structure by considering the subspaces of $C(X, \mathbb{C})$ consisting of elements with purely real resp. purely imaginary spectrum; $C(X, \mathbb{C})$ is always the direct sum of these, and the *-structure acts by the identity on the first bit and by $-1$ on the second bit.