1) Rabban Yochanan ben Zakai was almost certainly Nasi for some time after the destruction of the Second Temple. Per the Iggeret of Rav Shrira Gaon (written 987CE):
"ולא מצאנו אדם שנקרא "רבן" חוץ מן הנשיאים רבן גמליאל, רבן שמעון, רבן יוחנן בן זכאי, רבנו הקדוש."
My Translation:
and we haven't found anyone who gets the honorific "rabban" outside of people who were Nasi: Rabban Gamliel, Rabban Shimon, Rabban Yochanan ben Zakai...
2) On the question if Rabban Yochanan ben Zakkai was a priest, there some conflicting evidence.
תורמסי תרומה - הצריכות לשמרן בטהרה וגם הוא היה כהן כדאמרינן [בתוספתא דפרה] (פ"ג) אם מה שעשו ידי שכחת
My Translation:
And he [Rabban Yochanan] was also a priest as we see (in Tosefta) "what my hands did".
The Tosefta in question (מה שעשו ידי - "what my hands did") refers to a statement Rabban Yochanan makes where he claims that his hands forgot how to do certain duties that only priests were allowed to preform. Rashi reads from this that Rabban Yochanan was indeed a priest.
3) On the other hand, Tosefot have the opinion that Rabban Yochanan was not a priest
שהכהנים דורשין מקרא זה לעצמן. משמע דרבן יוחנן בן זכאי לא היה כהן והא דתניא בסיפרי גבי פרה שאמר רבן יוחנן בן זכאי לתלמידיו מה שעשו ושרתו ידי שכחתי לאו דהוא עצמו קאמר אלא כלומר ע"פ הוראתו
My translation:
The priests justify this text to their own benefit. This means that Rabban Yochanan be Zakai was not a priest. And as to what it says [in Sifrei about Parah] that Rabban Yochanan said to his students "what my hands did" is not that he did it himself, but it was according to his ruling.
Basically, Tosefot claims that since Rabban Yochanan rejects the priestly interpretation, this means he was not a priest. They further resolve the Tosefta ("my hands forgot") by saying that statement refers to a ruling Rabban Yochanan made about the priestly duties, not that he did them himself.
4) Related to your question, the Ramban on Niddah (49b) says:
והא דקתני בסנהדרין ואין הכל כשרין לדון דיני נפשות אלא כהנים לוים
וישראלים המשיאין לכהונה הא פריש רב יהודה דלמעוטי ממזר וגר אתי אבל חלל
אינו בכלל.
My translation would be roughly as follows:
"Jews whose daughters are allowed to marry Priests", is to exclude people born out of forbidden relationships and converts, but a Chalal [a Jew born out of forbidden relationship between a priest and a divorced women] is not excluded.
Thus, the above criteria allows pretty much any Jew "in good standing" to join the Sanhedrin. Thus regardless of Rabban Yochanan's origin, this in itself was not a legal reason to exclude him from serving as head of the Sanhedrin.
