"I think they have seen this movie, don't I?" --> Here if you are asking "dont I?" means "am I not thinking?" How can someone else know about your thoughts? [:^)] Hence in this type of scenario you must frame your tag question based on objective clause.
Similarly in this case, "He thinks we should come back soon, doesn't he?" --> How will "we" know "his" thoughts??? [:^)]
This makes sense. Then, can I learn it this way: if the main clause has the verbs meaning such as personal feelings, we can't ask tag questions based on the main clause. On the other hand, if the verbs in the main clause are action verbs, can we use it toe ask tag questions?
For example, People said the earth was flat before, didn't they? is it a correct sentence?
That's right. just one correction In the first clause the verb form "said" is not an action verb but a reporting verb. other examples of reporting verbs are: say tell promise etc...
Action verbs on the other hand include work, study, play, dance etc.