Yes they do
No they don't. It's been stated plenty of times in the series. They base their gender on their hosts, but Yeerks themselves don't have genders.
Book 19, Aftran explains the Yeerk reproductive process, including mentioning that they have 5 genders. None of them really correlate to male or female though, so those pronouns typically come from the bost.
If Yeerks have 5 genders, why are only 3 needed for reproduction?
Once I got the Sharing set up, I would invite the President to meet with Sharing leaders and plug a Yeerk into his head. Then the President could help me Yeerkify the other world leaders! This should make it easy to consolidate the planet.