Does the higgs field permeate all of space
Yes. All of the quantum fields describing elementary particles permeate all of space. What is different about the Higgs field is that it is nonzero (technically, it has a nonzero “expectation value”) everywhere.
and pull it this way and that based on its attraction to higgs bosons that reside within all matter?
No. The nonzero value of the Higgs field itself gives mass to quarks and charged leptons. Higgs bosons are not involved in this interaction. You can think of a Higgs boson as a ripple in the Higgs field. The Higgs field doesn’t need ripples to give particles mass. It just needs a uniform nonzero value. Higgs bosons do not reside within matter.
In this way, do concentrations of mass (of varying density) attract varying amounts of higgs field, in turn morphing varying amounts of space?
No. General Relativity does not require a Higgs field. Spacetime curvature depends on the density and flow of energy and momentum. Things with mass have energy, and when they move they have momentum. (But things without mass, like photons, also have energy and momentum.)
They show imagery of matter passing through the higgs field, experiencing greater resistance due to variations in mass.
This is just pop science. Matter particles don’t have “greater resistance due to variations in mass”. They have greater or lesser mass because they couple more strongly or less strongly to the nonzero Higgs field. The Standard Model doesn’t explain why their couplings to the Higgs field are different, resulting in their different masses. The various couplings are just parameters of the model.