A bit of research has convinced me the term "Mote series" is used, and is in fact more common than "Moties series". The search terms "Mote series" Jerry Pournelle yield 7200 results on Google, while "Moties series" Jerry Pournelle yield only 500-odd. Given that the terminology used by Wikipedia will actually skew results towards "Moties series" (since so many other websites copy Wikipedia's content, any phrase used on Wikipedia will generate hundreds of hits from that fact alone), I think this means that "Mote series" is by far the more popular term.
The main books in the Mote series are The Mote in God's Eye, The Gripping Hand, and Outies - the latter being an authorised sequel written by Jerry Pournelle's daughter and marketed as "Mote Series, Book 3" (it's even mentioned as such on the Wikipedia page you linked to).
Meanwhile, the term "CoDominion series" seems to refer to all works set within the imagined history that begins with an alliance (the CoDominium) between the USA and a revived USSR, who manage to colonise various planets before the alliance breaks down. Most of these books are written or co-written by Jerry Pournelle, but they don't form a single unified series, more a collection of works which are vaguely connected by being set within the same universe. As far as I can understand, though, the Mote series does form a coherent and unified narrative.
So, let's consider your bullet-pointed suggestions:
Do we need both mote-series and co-dominium-series, or should one be merged into the other?
Yes, let's keep them both. There aren't very many questions on either of them at the moment, but that could change, and there's plenty of precedent for having one tag for the wider series or universe and more tags for sub-collections of works within that.
However, it might be worth hunting around for more questions to tag with codominium-series. This tag is only used on one question so far, but there may be others (not necessarily mote-series questions) which deserve the tag.
No. However, when you write a tag wiki for mote-series (as I'm sure you plan to do shortly), it's worth mentioning that it can also be called the Moties series. I don't think there's any point in setting up a synonym either, since when people type
mot into the tag box they'll get this tag via autocomplete anyway.
Yes. I've just done this, and edited the tag wiki to say CoDominium instead of Co-Dominium.
EDIT: on the advice of user14111 (who's far more knowledgeable about 20th-century science fiction literature than I am), I renamed the tag to codominium-universe instead. Note that this required no editing of posts: no front page was harmed in the making of this answer.