Maybe the best way to understand the use of cases, is not to think in English, but try to look further to the big tree of Indo-european languages, with originally eight cases. In the monstruous sentence Look Brian, Pol gives the cat the food from the fridge with the help of his sister's spoon in his house in Waterford. All these parts of the sentence are in Sanskrit or Russian in different cases. All these cases have a function in the sentence and they were discernible in word endings. And they still are explicit in many other languages, but hardley in English. Brian is the Vocative or the Direct adress of the sentence. Pol is the subject, in the nominative case. The food is the direct object, in the accusative case. The cat is the one whoM (mmm, English case ending) the food is given to, the indirect object, in the dative case. From the fridge states the source or ground, the ablative case, in some languages the prepositional case. With the help of indicates the instrumental case. His sister's spoon indicates belonging or possession, the genitive case. In his house in Watford gives the location, the locative case. Then imagine you actually hear this different case endings yourself in real life. I left the allative case out (the direction or towards case). To hell! , or the more neutral downhill is a good example for that case.
"cat" is of the first declension
confirmed by the dictionary, which also shows its male...
rules for first declension are here
so far, "cat" nom. sing. has modified to "cat" gen. plural. Now final rule: all genitives plurals get eclipsed when you prefix with "na"
2.1 When do we eclipse nouns?
Collins Dictionaries (2011-07-28). Easy Learning Irish Grammar (Collins Easy Learning Irish) (Irish Edition) (Kindle Location 404). HarperCollins Publishers. Kindle Edition.
- 5 groups of eventualities
No, it is not The easiest way would be to look it up.
(If would want to deduct it from a set of rules, then you would have to start knowing the gender and then to go further from that. Event declension was not easy to figure out with the shown rules ...)
No, na gcat is plural and genitive. If it were the food of one cat, it would be bia an chait.
If you're wondering about the position of the apostrophe, it goes after the pluralizing "s" in the case of a plural possessive. "The cat's food" means "the food of one cat", whereas "the cats' food" is "the food of at least two cats."
One rule of thumb that usually helps me is that eclipse usually happens in the plural (like here), while singular nouns are more often lenited. That is not nearly a rule, but probably close to 2/3 of the time.
I definitely agree that it is confusing that cat can mean both "a cat" and "of cats."
I found out how it works here: https://en.m.wiktionary.org/wiki/Appendix:Irish_mutations. The presentation is quite clear, so you can scroll down to the thing you're currently trying to figure out and ignore the rest for now. Practical having the whole subject on one page, too, so you don't have to look in various places every time.