Where do you notice mobility restrictions?
Are you bracing the trunk when perforing the above movements?
When you were jumping rope it sounds like your right heel was not kissing the ground. Thr calf remained contracted the entire time. When you changed your focus there may have been a technique adjustment (conscious or unconscious) and your heel began kissing the ground.
When you run do your arms cross your midline?
Trunk rotation could contribute to the belt moving.
Are you doing movement/skills which incorporate all planes of movement?
Working primarily within one plane of movement the body can develop imbalances.
Incorporating movement within all planes of movements during warm up is key.
Have you watched the videos in the 14 Day ChallengeI 7 Green Lights which address the hip hinge archetype?