1. What is the definition of obedience and obedience of Allah & Prophits?2. What the meanings of sincerity in belief?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Obedience has two definitions, one linguistic and the other in Islamic terminology.

Linguistically, obedience means agreement and submission to comply with what someone in authority orders.

In Islamic terminology, obedience means to abide by the Orders of Allah and avoid His Prohibitions. Allah is The Creator, The Commander. So, all people should do what He orders. Allah Says (what means): {Unquestionably, His is the creation and the command; blessed is Allah, Lord of the worlds…} [Quran 7:54]

As for the obedience of the Prophet it means as Al-Qaadi Eyadh said: “The obedience of the Prophet means the adherence to his Sunnah and the full execution of what he established.' This obedience is compulsory on every pubescent Muslim just like the obedience of Allah. Allah Says (what means): {He who obeys the Messenger (Muhammad), has indeed obeyed Allah...} [Quran 4:80]

So, the obedience of the Prophet is part of the obedience of Allah as the Prophet is the reporter of what Allah revealed to him and all that he utters is but a revelation.

For your second question, know that sincerity is the opposite of polytheism. So, sincerity is the fact that the person abides by an order or avoids a sin just for the sake of Allah not seeking any praise from any body for such acts of worship. Allah Says about good Muslims (what means): {[Saying], "We feed you only for the countenance of Allah. We wish not from you reward or gratitude. Indeed, We fear from our Lord a Day austere and distressful."} [Quran 76:9-10]

Allah knows best.