1) I heard that there are certain traditions that mention that the prophet sees our deeds once or a couple of times even a week? So he can see our good deeds and our sins, is this true?
2) If the above is true, it is mentioned in Shabaniya Munajat:
إِذْ لَمْ تُظْهِرْهَا لِأَحَدٍ مِنْ عِبَادِكَ ٱلصَّالِحِينَ
«You have not revealed my sins even before any of Your righteous servants». So wouldnt the prophet as well count in those righteous servants, or can he see the deeds since he is our prophet?

1- Yes we have numerous hadiths that the Prophet (s) sees our deeds/sins. For example, in the book of Kafi, volume 1 page 171 we find this hadith:

عن علي بن إبراهيم، عن أبيه، عن عثمان بن عيسى، عن سماعة، عن أبي عبدالله (عليه السلام) قال: سمعته يقول: ما لكم تسوءون رسول الله (صلى الله عليه وآله): فقال له رجل: كيف نسوءه؟ فقال: أما تعلمون أن أعمالكم تعرض عليه، فإذا رأى فيها معصية ساءه ذلك، فلا تسوءوا رسول الله (صلى الله عليه وآله) وسروه.

Also, Sura at-Tawbah verse 105 indicates he sees our actions.

2- As for the Du’a you mentioned, we can say that the Prophet (s) is an exception and that the righteous servants don’t include him. Or it’s possible that some sins only Allah sees. Allah knows best