What is Buddhism's view about debating teachings? What is Buddhism's view about debating teachings: should we accept teachings without question, or should we have our own internal and social debates about what we learn?

If we should debate, what is the correct way of doing it?
Another way to put your question (re-phrase) is... "Is criticizing other religions allowed in Buddhism?"

If that criticism is fair and aims to correct others, then it’s allowed and is a good thing to do. As it is mentioned above in Cula-dukkhakkhandha Sutta, Jainism is criticized by Supreme Buddha.

Supreme Buddha was one of the first crusaders against caste system, an apartheid evil ideology started by Aryans who used it to suppress indigenous Indian people. Lord Buddha was a social reformer and thoroughly criticized the animal sacrifices and discrimination of women which was common in the Hindu society.

A heartfelt touching discourse of Lord Buddha on animal sacrifice comes in the Sutta Nipata, Brahmana-dhammika Sutta. In this discourse, it is discussed about the ethical conduct fit for a Brahmin. Lord Buddha speaks respectfully of ancient Brahmins who rejected disdainfully the taking of life and never allowed their religious rites to be contaminated by the blood of poor animals. But corruption came upon, and they started the wicked practice of animal sacrifices to please gods. When the knife was laid on the neck of cow, the gods themselves cried out in horror, seeing that crime of ingratitude and insensitivity committed on an animal that was such a faithful worker, such a sustain-er of life, to humans.

In the Agganya Sutta, Supreme Buddha focuses on the Brahman idea from Wrig Veda that Brahmans came through the mouth of Maha Brahma/God; thus they are supreme. Lord Buddha questions the Brahmins as to how they can claim so when they see day to day that women in the Brahmin caste get pregnant, give birth, and take care of their children? Don’t they understand that they all came from the mother’s womb, not from the Brahma’s mouth?

‘One is not low because of birth nor does birth make one holy. Deeds alone make one low, deeds alone make one holy.’
– Supreme Buddha (Dhammapadha)