Does that mean that when two Quarks come as close to each other as the Planck distance 10^{-33}m then gravity must be taken account if one is to understand their movement?

I know vaguely that the Planck energy is the point at which a black hole forms or something like that but I'm not really sure. Any advice would help out.

The Planck energy is not directly relevant to the formation of black holes - the latter is governed primarily by the Schwarzschild radius of the quantity of mass under consideration.

More or less. Bearing in mind that what really happens at the Planck scale is sheer speculation....

If you confine a particle in a smaller and smaller region, its zero point energy increases. The Compton wavelength is the distance scale ~ħ/mc at which the zero point energy equals the rest mass, and therefore particle number is no longer conserved.

If you confine a mass in a smaller and smaller region, its gravitational potential energy increases. The Schwarzschild radius is the distance scale ~Gm/c^{2} at which the gravitational potential equals the rest mass.

The Planck scale (Planck distance, Planck mass, Planck energy) is where quantum mechanics and gravitation become equally important, i.e. when the Schwarzschild radius equals the Compton wavelength.