Etze li hashuka, אצא לי השוקה (the market song, a modern remake of the kids song, with the same lyrics, but for adults) has all the animals in this skill pronounced. https://youtu.be/kXNXH2oZRRs (The link to the lyrics and transliteration are in the video's comments) Plus it's an awesome song (she also does a great modern alef bet for adults).
No, only rooster. Hen or chicken are תרנגולת, or if you mean chicken in the sense of food, then it's עוף.
A rooster (cockerel) is a male chicken, so why isn't chicken and chicken accepted the same way other animals are, the sex is only relevant for agriculture...
After the "and" it does. https://ell.stackexchange.com/questions/136237/article-before-the-noun-before-and-after-and
If you are asked, what are you eating? (You can say: chicken)... If you are asked what is that in the yard, it would be: a chicken.