Your first misconception is the thing about kinetic energy of particles: Particle energies follow a distribution that does not have a cutoff. I.e. there will always be particles present with enough energy to go into the gaseous phase. If you lower the temperature, the count of these particles decreases, but it never reaches zero. Consequently, you get sublimation whenever you have an ice surface exposed to perfect vacuum. The count of sufficiently energetic particles dictates the maximum rate of the sublimation, and there is always a vapor pressure at which this rate of sublimation equals the rate at which gaseous particles hit the ice, dissipate their energy, and enter its phase.
The melting point is largely invariant of pressure because melting/freezing does not change volume much. Volume grows a bit when freezing at room temperatures, so very high pressures can discourage water from entering the solid state, lowering the melting point a bit.
Of course, the same volume dependency is present with the gaseous phase: Water takes much more space when it's low-pressure vapor than when it's solid or fluid. As such, it's natural that both the sublimation and the boiling point show a strong pressure dependence: The more pressure, the more energy (= temperature) you need to put a particle into the gaseous phase. The volume dependency between vapor and other phases is much, much stronger at the triple point than the volume dependency between solid and fluid phases, so you have to expect the sublimation point to qualitatively continue the path of the boiling point to lower temperatures and pressures. There is a slight angle between the two curves at the triple point, though, as far as I remember.