Is "kyckling" pronounced as "shick-ling"?
Almost :). You can find two more pronunciations of kyckling here.
Is the 'y' in Swedish more like a rounded version of the Swedish 'i'? It sounds like the place of articulation is about the same, but the enunciation sounds more muffled which suggests a rounded vowel?
Late reply, but yes, it’s exactly the same as i but with rounded protruded lips.
Do these refer to the animals themselves as well as the cooked dishes, or is there a different word for the live animal?
This sentence refers to food I guess, since else you would say "en fisk" or "fiskar" and "en kyckling" or "kycklingar".