Learn English – A word for someone trying to make people underestimate him

I don't believe that there is a single word which describes what you're asking for. You probably need a short phrase to give this idea:

He expects others to follow his example.

He wants his son to take after him.

He thinks he knows what's best for everyone.

There are a lot of words that mean "having an extremely strong opinion." Here are some good ones.

Ardent

Passionate

Zealous

Fervent

Vehement

Also, Partisan. Someone who is so strongly in favor of a particular cause of another is called partisan. This is generally used in politics to describe members of political parties who only act in their party's best interests.

For the second question, a word that describes a situation where one can only believe one way or the other:

Controversial

Polarizing