English Clauses (Grammar)

Hi there! I have few questions. I will really appreciate it if someone could explain all these to me.

1. ''Some part of me needed to hear you say that running away together wasn't an option'' is a matrix clause, which means main + subordinate.
Is ''that running away together wasn't an option'' an object complement?

2. ''He thinks that his biggest mistake is what he did yesterday.'' What kind of clause is this?
I mean, ''He thinks that his biggest mistake is what he did yesterday.'' - Noun as Direct Object?
''...is what he did yesterday.'' - Noun as Subject Complement?

3. ''She is not the doctor that I am looking for.'' Is this a Subject Complement or relative pronoun? ''She is not the doctor'' - Subject Complement?
''that I am looking for'' Relative pronoun?

4. Can a relative pronoun uses as complement? Just like No 1 and No 3.

Thanks in advance.