Se is a pronoun. It is used as direct and indirect object designating the same person or thing as the subject in verbs at the third person. To designate something else, use le, la, les, the speaker use me, nous and the person(s) to whom one speaks, use te, vous.
Then there is the class of pronominal verbs for which the use of a reflexive pronoun (se at the third person, me, nous at the first, te, vous at the second) changes the meaning and for which the function of the reflexive pronoun is not well defined. Trying to analyze them is doomed for the same kind of partial success as trying to analyze phrasal verbs in English by separating the verb and the preposition.
In il se lave we are clearly in the first situation.
In Paul se dit..., we are perhaps in a case where a verb is becoming pronominal, you better analyze that as Paul dit à lui-même..., but it is acquiring a slightly different meaning (and different from the impersonal use of il se dit).
In il s'appelle Paul, the verb has become pronominal in this use (he is named Paul is a better translation than he calls himself Paul) but you can somewhat get the meaning from a simple analysis.
In il s'ennuie, the meaning is clearly different (he is bored, not he is annoying himself).