Would impeachment have removed president Nixon from office?

Not necessarily.

Explanation:

Impeachment does not remove the president from office in of itself, although that is often a step that is taken after impeachment.

freedom of speech is also protected by the 1 amendment.

explanation:

here is pretty much what the 1st amendment says:

congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances.