Which comes first and takes precedence in the sight of Allah Almighty: rights or duties? And does the order differ between one's dealings with the Creator and one's dealings with creation? And is there a general rule or elaboration regarding this?

Regarding one's dealings with people, one begins with one's duties, and this does not prevent one from demanding one's rights. If the two conflict, one should start by fulfilling one's duty. As for dealing with the Creator, the creature has no right over Allah the Almighty. What is mentioned in the Hadith: "Indeed, the right of Allah upon His servants is that they worship Him and associate nothing with Him, and the right of the servants upon Allah is that He does not punish one who associates nothing with Him," this is a right that Allah has made incumbent upon Himself out of generosity and grace. The creature has no entitlement over Allah as one creature might have over another. And excessive questioning is blameworthy in Islamic law.