(1) No, the Hodge index theorem, valid in arbitrary characteristic, is the Hodge standard conjecture for divisors on surfaces. Taking 2-dimensional linear sections, one can deduce the Hodge standard conjecture for divisors on arbitrary smooth projective varieties.
The Hodge standard conjecture over $\mathbf{C}$ is not a consequence of the Hodge index theorem but rather of the fact that the space of algebraic cycles is contained in the space of Hodge classes $H^{p,p}(X)\cap H^{2p}(X, \mathbf{Q})$, and by Hodge theory (Riemann bilinear relations), the Lefschetz pairing is definite on the primitive parts of those a priori bigger spaces.
(2) I do not think there is an algebraic proof, since the above arguments requires some "positivity" (a definite pairing on a $\mathbf{Q}$-vector space remains definite after restricting to any subspace).
Speculation: Maybe the Weil cohomology theory for varieties over $\overline{\mathbf{F}}_p$ with values in the Kottwitz category ${\rm Kt}_{\mathbf{R}}$ whose existence was recently conjectured by Scholze could make such an argument possible in positive characteristic.
(3) Tetsushi Ito in "Weight-monodromy conjecture for $p$-adically uniformized varieties" (Invent. Math. 2005) crucially proved the Hodge standard conjecture for the varieties obtained from $\mathbf{P}^n$ over $\mathbf{F}_q$ by successively blowing up all $\mathbf{F}_q$-points, the strict transforms of the lines between those points etc. The cohomology of these varieties is generated by algebraic cycles, but they do not lift to characteristic zero.
(Obviously, if a variety in characteristic $p$ whose cohomology is generated by algebraic cycles lifts to characteristic zero "together with all cycles", then the Hodge standard conjecture holds.)