PARCHMENT WARNING: watch out, this is really long and I'm terribly sorry for it.
"Loro" derives from "illorum", genitive plural of "ille" (masculin). It means "that", but it can also be used as a pronoun to address something or someone.
Anyway, the other Italian possessives derived from their Latin counterparts, which were variable in gender and number (and in cases, but that's a story for another time including unlucky Italian students forced to learn declension tables and other Latin horrors. Ty Ministry of Education). "Illorum" is plural, sure, but it had the feminine counterpart "illarum". The thing is, having "loro", "laro", "liro" and "lero" would have been extremely awkward and foreign to Italian grammar prescribing the vowel replacement to happen only at the end of the word and not inside, so I really think this quirk (I'd call it one more of an irregularity in this cursed, messy but beautiful language) is due to the very proven concepts of force of habit and the unspoken (but familiar to any human with linguistical abilities) respect of pre existing grammar. Anyway, the evolution period of Latin becoming 1300 Dante's Florentine dialect stretched all the bloody Middle Ages and therefore is hella obscure and mine's just a hypothesis, so take with a grain of salt.
Out of curiosity, in Latin "loro" was "suus" (and sua and suum, this is the singular). It was used eithed for the 3rd person singular AND plural. Going back at the obscurity of that period I personally dislike called Middle Ages, I really don't know what was that made people restrict "suus" only to 3rd person singular. Maybe they just wanted to distinguish between what belonged to a person and what to a crowd.
If you have found out something, please tell me under this long comment.
Hope I've helped (and been clear to the extent of English being my L2 allows me)!
For "their" it's always "loro" but it has to have a definite article preceeding it that corresponds with the object that is "theirs." So, it's "la loro" here because "cena" is feminine singular, and "la" is the article for that. If it was "Their dog eats" It'd be "il loro cane mangia" Same for plural. For example, "the women love their children" - " Le donne amano i loro bambini"
When used as an ADJECTIVE, they all mean "own" (e.g. his [own] apple; her [own] cats. Which version to use is dependent on the NOUN as the adjective needs to match in gender and number.
Proprio - Masculine single / Propria - Feminine single
Propri - Masculine plural / Proprie - Feminine plural <-- remember this is when the noun is plural, not the people 'owning' that noun
Proprio is also used as an ADVERB to mean "really". Sono proprio stanco = I'm really tired. When used as an adverb, the ending doesn't change; it's always "proprio".
You'd be understood but it's that kind of strange that would leave you with a philosophical question, is it in or on? Jokes asides, in a reverse comparison, if someone said "your dinner is in the plate", you'd understand him as well but would feel that something's a little off, despite it not being a grammar tragedy.