Will a Muslim be held accountable for what occurs within himself, such as du'a (supplication) or the two Shahadas (declarations of faith), given that Allah does not hold one accountable for what is not uttered or acted upon?

There is no accountability for what goes on within the soul unless it is uttered or acted upon. A supplication (du'a) is not considered a supplication merely by intention, but by uttering it. As for the two testimonies (Shahadatayn), they must be uttered, along with understanding their meaning and acting upon their implications outwardly and inwardly.