1982 is a dog and 1977 is a snake. Do they have conflicting personalities? Can they come together?

The question you asked is not correct. There is no conflict between personalities, only that personality is inappropriate. In terms of the zodiac, dogs and dragons are in conflict, and snakes and pigs are in conflict. Your zodiac is fine. However, if your horoscopes match, you will know according to your date and time of birth.