Hi OneHipDude, interesting situation you find yourself in!
Most hippies who run were runners, and I vaguely remember reading a couple of surgeons advising that runners could go back to running but non-runners should not contemplate starting... I started running age 30 and am still running age 70 despite a BHR and a THR, in both cases I was jogging again within 8 weeks, and I never really warmed up to speak of although I have recently started adding a 20 minute jog/sprint warmup before 5km races and think it helps a bit. Corey's article is pretty good, I just re-read it, and advice like landing on your mid-foot placed directly below your COG is correct - lean forward slightly from the ankle - and shorter steps with a higher cadence will minimise stress on your hips. I tended to return to running using unpadded racing shoes to stop me heel-striking, it seemed to work to focus me on quick pattering steps.
You don't say how much the treadmill plan has you doing, or how quickly the effort ramps up, but quitting when pain starts is the correct response. Do you have a history of long, fast walking on your new hip or is the walk/jog/run experience completely new to you? If the plan is too ambitious in increasing effort then adding more rest days or repeating weeks might be a solution. The pain you have had will almost certainly be scar tissue rather than device related at this time from your op.
Diet-wise I have no advice other than limit processed food and seed oils, no idea if synovial fluid production is related to diet, decent hydration is important though.
Good luck!