How does coupling to the Higgs field take a particle with a spacetime interval of zero and give it a non-zero value, i.e, how does it move it off the null geodesic? [of course, the Higgs field is responsible for a fundamental particle's rest mass, assuming a non-zero coupling constant, and such particles are confined to time-like intervals, okay, but why?]
First, note that there is no unified theory of QFT and gravity, so talking about geodesics and about the Higgs is really not possible within the framework of our current theories.
Nevertheless, the confusion here seems to stem somehow from the idea that all particles are "initally" massless, and "then" the Higgs comes along and gives them mass. This idea of "inital masslessness" is ill-defined. The Higgs field exists, and so the other particles have mass (if the Higgs has acquired its non-zero, symmetry-breaking vacuum expectation value). There's no movement occuring, the difference is discrete - if the Higgs has vanishing VEV, particles are massless, if it has non-vanishing VEV, particles have mass. It is a sharp phase transition, not a smooth process.