Q: If you see a Muslim doing kufr by mocking Islam as an example or doing something else that is kufr, then he later does the 2nd kalima, perhaps for wudhu, but then gets informed that what they did in the past was kufr, is it fine if they just repent without doing the 2nd kalima again? If they don't, are they considered as disbelievers?

A: They should renew their imaan by repeating the kalimah. Similarly, they should make sincere tawba for the wrong statements they made.
However, if they do not repeat the kalima to enter into the fold of Islam again but they continue to perform salaah, make zikr or recite the Quraan Majeed, then the kalima is found in these ibaadaat, their imaan will be renewed and they will once again enter into the fold of Islam.
And Allah Ta'ala (الله تعالى) knows best.
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