Which prayers should be started with in qadaa' (making up for missed prayers): should it be the prayers missed at the beginning of puberty, or those missed recently, or those missed due to an excuse first, then those missed without an excuse?

The person afflicted with waswas (obsessive doubts) should adopt the easiest and lightest of opinions. Therefore, there is no harm for you in acting upon the opinion of Shaykh al-Islam and those who agreed with him, that there is no obligation to make up prayers deliberately missed. Alternatively, you may act upon the opinion of the majority (Jumhur) that making up the prayers is obligatory, and you may begin with whichever prayers you wish. For the Shafi'is, the sequence among missed prayers is not obligatory, and a person afflicted with waswas should not follow the stricter rulings of the madhhabs.