Interlocutor is a word that is rarely used in ordinary conversations.
What are the best ways to ask interlocutor whether he understands you in different circumstances (formal conversation, informal talk)?
It depends on the circumstances, an interviewer might take a more authoritative tone, an interviewee might choose a more submissive or respectful tone
Formal: "Am I being clear?"
Informal: "Does that make sense?" or "Yes?"
What are the best ways to answer such questions meaning 1) I understand, 2) I understand and agree, 3) I don't understand.
There are many ways and what is "best" depends on context - relationship, mood, circumstances.
1) "I understand."
3) "Pardon?" (informal, polite) or "what on earth are you talking about?" (informal)
"Sorry, I don't understand, could you please explain." (more formal, respectful) or ...
What are the best ways to let speaker know that you understand him and/or agree or don't understand when you was not asked?
Say "I don't understand you", "I agree" or "I don't understand". In the first or last case, simply insert "Sorry, " at the start to increase the politeness level.