God's algorithm is not computationally feasible on consumer hardware so I'd assume not although there are many algorithms that can get pretty close (either matching or 1-2 moves off the optimal solution) which are much faster to solve. If you're curious, look up Cube Explorer which is an app that's built for this.

Not in practice. Computing the absolute minimum solution for every possible position is computationally infeasible for a web-based solver. This uses Kociemba’s two-phase algorithm instead, which produces very efficient solutions, usually close to optimal, without requiring enormous amounts of time and memory.