How can one deal with the problem of frequent absentmindedness during prayer and forgetting the number of rak'ahs? And is what happens considered a form of waswas (whispers from Shaytan) or mere absentmindedness, especially given the recitation of Quranic verses in the wrong place, along with an inability to firmly memorize the Quran?

Humility (khushu') in prayer is its essence and spirit. Prayer does not need to be repeated if humility in it is diminished or if the mind wanders. If doubts become numerous, it is better to ignore them and not pay attention to them. If you doubt whether you added an extra unit (rak'ah) or not, assume that you did not add it. As for continuing the prayer after realizing that you are in the third unit, there is no harm in that. You should ignore doubts if they become numerous and strive to achieve humility. If you are cured of excessive doubts, and then a doubt arises, then build upon the lesser number.