The latter is a "smikhut" (סמיכות) form. That means the word is part of an expression. For instance, "meal" is ארוחה, but "breakfast" is ארוחת בוקר, literally "a meal of the morning".
The plural of ארוחה is ארוחות. Note the additional Vav.
"I eat a meal" is a literal translation from Hebrew, but it is not English. It should be "I have a meal" (or maybe "I eat flour"?). Even if you say "I have a meal", the sentence will sound incomplete, because you would expect it to be continued for example, by saing "I have a meal three times a day" or "I have a meal every now and then".
I disagree—I think this is fine English. A little odd with no context, but most Duolingo sentences are.
Sure! It might be something like: "I come home, then I eat a meal" or "When I am hungry, I eat a meal." But I do agree with you that it sounds a bit more natural to say "I have a meal."
In your examples, it's part of a longer sentence, but I can't imagine it being used on its own.
You're telling someone about what you do every day: "I go to work. I come home. I eat a meal. I go to bed." Or it could be a response to a question. "What do you do when you're hungry?" "I eat a meal."