So that when the uncertainty in momentum Δp tends to zero, the uncertainty in the position Δx tends to infinity (a particle with a perfectly defined momentum is a plane wave which is everywhere).

Then how would an object move if it had to travel along an infinite amount of points?

This would be indeed physically problematic in a classical continuous model, but this whole picture has changed with quantum mechanics, where the position of a moving particle is undetermined and smeared into a probabilistic cloud. Moreover, if spatial coordinates are equally probabilistically unsharp, a continuous space may be defined which only allows for a finite number of degrees of freedom.

Space being continuous has everything to do with the existence of a minimum lenght

This is not a necessary condition in quantum theory, eg in Wheeler-DeWitt cosmology the fundamental Planck length coexists with continuous space and time. This point should be taken with great care: space is no longer assumed classical, and as such can’t be realistically modeled with a sort of “chessboard” where particles would jump from Planck Length 1 to Planck Length 2 and so on. Space should be considered quantum before questioning its continuity or discreteness. First, there is an uncertainty in the position of particles, making their location in the “chessboard” fuzzy and unsharp. Second, quantumness of spacetime should imply that the “chessboard” is itself fuzzy and unsharp. Third, spacetime could not be a fundamental entity, and discrete properties of its underlining structure (not characterized by lengths or times, but still defining a geometry) should not automatically imply that such spacetime is lacking continuous properties. Fourth, some models in topological field theory show both discrete and continuous dual characteristics, so that some deep connections may well exist physically between these apparently opposed notions.