Hmmmm, my first thought is tight hip flexors and weak/inhibited glutes. Here's the thought process: lower back tired while jogging is due to anterior pelvic tilt from tight hip flexors forcing you to use your lower back to keep your torso upright, which gives your lower back a lot of work making it stronger (explaining some why you are strong on hypers), lower back is fine during deadlifts because it isn't hyperextended, squats ending in good mornings could be a lot of things, but it's likely that out of the hole you shoot your butt back and the knees come in because your glutes are weak so you are shifting weight to what is stronger (hams and low back). Of course that is all conjecture without seeing vids of you squatting and deadlifting.
SO POST SOME VIDEO IF YOU WANT HELP!
But if what I said is true the fix isn't that hard: stretch hip flexors and activate and strengthen the glutes and everything will get better including running/jogging.