Is it permissible to designate supplications for specific parts of the prayer as a form of arrangement, such that the worshiper intends to supplicate with such-and-such in the first rak'ah, and such-and-such in the second, and such-and-such in the tashahhud, in order to remember the supplications and not forget them?

Supplication (Du'a) is recommended during prostration (sujud) and after the final Tashahhud. This is based on the Prophet's (peace be upon him) saying: "The closest a servant is to his Lord is while he is prostrating, so increase your supplication (du'a)." And his saying: "Strive hard in supplication, for it is worthy of being answered for you." The Prophet (peace be upon him) also gave the worshiper the choice to supplicate for whatever he wished after the Tashahhud. There is no harm in recalling what one wishes to supplicate for before or during the prayer, and distributing it across the rak'ahs (units of prayer) without believing in its sunnah (prophetic tradition) status or consistently doing so, but rather to aid in remembrance. It is best for the worshiper to supplicate with whatever Allah opens for him each time, and to make up for what he misses in other times of answered supplication.