Will God punish us for actions we consider bid‘ah (innovation), such as saying "Truly there is no god but Allah," or "Alhamdulillah (praise be to Allah)" after burping, or reciting Surat Al-Baqarah with the intention of having a need fulfilled, while others consider these to be righteous deeds?

If it is established that a certain act is an innovation (bid‘ah), then it is not permissible to say that it is a righteous deed. For religion is what Allah has legislated, and no one has the right to add to what the Prophet, peace and blessings be upon him, has brought. Allah has perfected the religion, and the Prophet, peace and blessings be upon him, informed us that "whoever introduces into this matter of ours that which is not from it, it is rejected." Acts of worship are based on divine revelation (Tawqeef) and following (ittibā‘), and refraining from innovation (ibtida‘).

As for the matters mentioned in the question: Saying "Truly, there is no god but Allah": This is a remembrance of Allah Almighty. It may be an additive innovation (bid‘ah idāfiyyah) if it is regularly practiced in a place where its regular practice is not legislated. Saying Alhamdulillah (Praise be to Allah) after burping: The words of some scholars indicate its permissibility, and it was mentioned that there is no known Sunnah for it; rather, it is a customary practice. Reciting a portion of the Qur’an with the intention of facilitating a specific matter: This is a form of seeking intercession through righteous deeds, and it is permissible.