As LazerA points out very nicely, the word in question does not necessarily mean "Dualist" However, if you are reading the verse with that translation, then the following would be the answer to your question.
1) Dualist is a person who believes that there are two forces in the world, the Good and the Bad and that they are independent of each other. This is in contrast to the Jewish belief that Gd Creates both Good and Bad.
(2) Sometimes, you there is such a large divide between philosophies, that merely being around each other will not bridge that divide. With Dualists, there is nothing you will be able to do to convince them that really all comes from one Source.
(3) Yes. There are two distinct authorities, Gd and Just governments.. however, Gd is also Lord and King.
(4) When it comes to Dualists, there is a danger in mingling with them. The danger, is that since they believe there are two major forces in the world, one good and one bad, they can always attribute any action to either the Good force or the Evil force. This creates a relativism that can lead to bad character traits. When a person recognizes that everything in life comes from Gd, and therefore, even the Evil in the world has a purpose and will ultimately come to some good, it greatly changes a person's perspective. It means they will try to make the best out of a situation, and any successes they attribute to Gd, and any failures they will attribute to Gd's will as well. Since you could never convince a Dualist of this possibility, hanging out with them will just cause you trouble. However, since the pasuk references "my son", it's very possible that this advice is only given to the average person, and not to someone who is prepared and well learned, and guarded against the negative influences of a Dualists' perspective in life.
There is a lot lacking in my answer, but hopefully it will be a springboard to further study regarding how Jewish tradition views followers of Dualist philosophies, and explain why there is a desire to avoid Zugim (pairs) in many instances.