Is it permissible to pray on behalf of the deceased, or is it limited to supplicating for them, giving charity on their behalf, and fasting on their behalf? And is what is known as "al-Shadir" (reciting the Quran in public places) considered an innovation (bid'ah)?

No one prays on behalf of another, because prayer is a purely physical act of worship. However, some scholars have permitted donating the reward of supererogatory prayers to the deceased. The best and most complete approach is to restrict oneself to what is mentioned in the Sunnah (definition): such as supplicating for the deceased, giving charity, fasting on their behalf if they had an obligatory fast due, and performing Hajj on their behalf if they had an obligatory Hajj due. As for setting up marquees to recite the Qur’an for the soul of the deceased, this is an innovation.