"Есть" is more "to be", even though the Russian way of phrasing possession uses it, implied and explicit. "У меня есть" literally means "at my [place] [there] is".
Would "кошка ест курица" not also work? In what situation would an Accusative be more appropriate?
Think of an Accusative form as of an English noun used as an object without any preposition: "I bought a car", "I killed a dragon", "I will send a reply shortly", "I cook salads", "I write books". This will not match Russian all the time but it will give you an idea. A so-called "direct" object of an action uses Accusative. However, this does not help you much because you cannot know beforehand which verbs have direct objects—you'd have to be a native speaker of a language to do that.
Still, "simple" verbs like "eat", "drink", "make", "write", "want", "see", "buy", "sell", "take" WILL interpret the object they operate on as a "direct" object, i.e. something seriously and directly affected by the action.
If you look at the list again, you might notice that "want" and "see" do not really belong there because their object is perceived by and agent but not actually affected by the action. Sadly, this is true; also, this is the reason verbs like "to like", "to listen" etc. may or may not take Accusative, and you cannot predict it. The use does not match up across different languages, and different verbs of the same language (with close meanings) may use different government.
On a brighter note, verbs like "to want", "to read", "to see", "to hear", "to know" DO take Accusative objects (such verbs are called transitive), and somehow this behaviour is quite common across languages—at least, European ones (but not only them, because they get similar treatment in Japanese).
The translation in english seems to be "the cat is eating the hen" but could it be "a cat is eating a hen"?
Y is accusative it makes it the object of the sentence Without it, both are subjects. So either the cat is eating the chicken or the chicken is eating the cat. So y is very important to show the chicken is being eaten. Also in Russian курица is generally used to mean chicken as in the food regardless of male vs female
Is it more common to call a chicken by the correct sex in Russian? In English you would use chicken most of the time when referring to a farm bird we eat. Rooster and hen are used but I wouldn't say they're regularly used in place of chicken.
What form of accusative does meat acquire as a direct object - animate or inanimate?
Here it makes no difference, since курицу is feminine, and the singular forms are the same. The plural forms are different, though, and I imagine there are some masculine foods for which being eaten alive would be unwanted - if that's a possible interpretation of the animate/inanimate distinction.
Such nouns are treated as animate except with шпроты (for some reason), and optionally with креветки, мидии, устрицы. A number of dictionaries also include words such as краб, кальмар, лобстер, омар.
Шпроты, however, are now more like the name of the dish because they are not always sprats.
Basically, these are words for some species of fish and small sea/freshwater animals that had not been much known as living animals prior to being introduced as "exotic" foods. So the well-known рак (crayfish) is always animate. To my ear, only шпроты, устрицы, мидии, креветки sound OK as inanimate, but then again, who am I to judge (I did not even eat fish before I turned 20).
Of course, it ONLY applies when you are talking about food.
Since кошка ест курицу means "The cat is eating THE hen/chicken(the bird)"
Could кошка ест курица mean "The cat is eating chicken(the meat)?