Guideline: I am not a linguist, and I struggle to use a smattering of Spanish and Russian in some character's dialog, without embarrassing myself. I will not, in fact, be inventing multiple languages for the various races.
That being said, I find it useful to make some of the names of things I come up with come from other race's languages, just for flavor. That's the closest I'll come.
Languages in RFI are as diverse as clothing, however, everyone needs to be able to communicate with each other. Because of that, almost everyone in the universe knows one or more of a handful of common languages.
English is considered a "universal" language, as the League (and to a lesser extent the Terrans) have popularized it's use across most of Human controlled space. Since the NorAellians and Lyndri have to trade with Humans, most have had to learn to speak it.
This is the accepted language of the Terrans. It is the only language spoken in schools until the age of six, then it's required that all students become bilingual with English.
Outside of Terran territories, it's rarely used, as all Terrans are required to learn English and it's the primary language of the League already, so why learn more than one? Some merchants who only operate in Terran space do, however, as a way of setting themselves apart of their competition.
Russian is still spoken in the former Russian territories on Earth, as well as very common in Freelance space. (The Russian territories are allowed an excemption to learning Mandarin from birth; they are taught Russian, and then Mandarin and English at age six.
NorAellians share one common language, which itself has no name. The idea of naming that-thing-that-everyone-must-know-in-order-to-communicate-and-understand-the-concept-of-naming-to-begin-with seemed silly to them. Their language focuses on taking simpler words or concepts and combining them into larger, more complex concepts. Their language is about efficiency; if you can express an entire setence in a single word, why wouldn't you?
This is the official Lyndri language. It's very rarely used outside of Lyndri space, or amongst Lyndri themselves. Rarely, it's referred to as "King's Tongue", but that usage is seen as being a bit revisionist and overly politically correct.