I think that "he eats a dog" would be "Itheann sé an madra", but I agree that the unusual (for me) word order makes things confusing.
I first thought the translation was "I eat the dog", because, to me "Itheann" and "Ithim" sound the same. Is there any difference in pronunciation between these two words?
Just like English, Irish makes a distinction between "eats" and "is eating." The present tense in German or French doesn't make that distinction, but here you have on the one hand "Itheann an madra" for "The dog eats", and on the other "Tá an madra ag ithe" to emphasize that the action is taking place at the same time as the speaking of the sentence: "The dog is eating." I see you're also doing Dutch: does that language make the distinction?