Is it permissible to supplicate with "O Allah, I seek refuge in You from a supplication that is not heard" and "O Allah, do not let the good deeds of the doers of good be forgotten"? Does the first supplication contradict Allah's attribute of Hearing, and does it show ill manners towards Allah?

The first supplication is a confirmed prophetic wording, and its meaning is to seek refuge from a supplication that is not answered. As for the second supplication, it has not been found as an authentic tradition, though its meaning is sound; because when forgetfulness is attributed to Allah Almighty, it means abandonment, not human forgetfulness. So, its intent is to ask Allah Almighty to fulfill His promise, "Indeed, Allah does not allow to be lost the reward of the good-doers" (Quran 9:120). However, it is better for a Muslim to seek out the supplications from the Quran and Sunnah and suffice with them.