It probably has to do with which language the words originally came in from. By the way the only one of these where it would feel completely wrong to use -kött is for fish. You just don't say fiskkött. You can often say just nöt, fläsk, lyckling and fisk without adding - kött, if it's in a situation where it's understood that you're talking about meat/food.
"Nöt" on its own actually means "nut", as in for example walnuts, but a Swedish-speaker isn't likely to confuse the two since I don't know if "nöt" can be used on its own to mean meat, and even if it can't I'm pretty sure it'd be conjugated differently than "nöt" meaning nut (which is an en-verb, I'm not sure now the meat meaning would be conjugated). "Nötkött" comes from "nötkreatur" which basically means "cattle" and I unfortunately don't know the etymology behind.
I guess you have a problem with the "k" in "kött" and "kyckling". ("Fläsk" is pronounced flesk, so that is easy). The "k" in "kött" and "kyckling" is close to the "sh"-sound, but not exactly the same. You can listen to natives pronouncing those words here:
What's what? Voiceless alveolar-palatal fricative? Voiceless - not vibrating the vocal cord - this is known as the phonation; alveolar-palatal - the roof of your month towards mid-front - this is known as point of articulation; fricative - forcing air out of your month when pronouncing a sound - this is known as manner of articulation. These are all important tools when learning to pronounce sounds from a different language that don't exist in your native tongue. I recommend you to look up and read the 'articulatory phonetics' article in Wikipedia to familiarise yourself with these tools.