Finally a question I may be able to answer! I have a Bachelor's in English Literature & Culture and have taken some courses on Old English and Middle English (mods if you want more verification feel free to pm), currently studying literature more so than (historical) linguistics, however.

Your question of how cow “became” beef is interesting, as the situation surrounding it is so illustrative of the distinctions between social strata in England throughout the centuries. The question of “when” is probably between 1066 (following the Battle of Hastings) and 1500. Prior to 1066, people in England (to whatever extent it could be called England at that time), spoke a mixture of Old English and Scandinavian. Many words with Scandinavian roots seem so normal to us now it sometimes comes as a surprise to learn of this ostensibly 'foreign' root. This, as Elly van Gelderen claims in her A History of the English Language (2006), demonstrates the close contact between Old English and Scandinavian speakers (see page 9 of said book, where she includes some examples such as call, crave, screech and most importantly in my opinion, egg). Van Gelderen highlights especially how “everyday” the shared words are. The close contact came about due to the frequent invasions by Scandinavian raiding parties, and the later occupation of areas such as Northumbria, in northern England. In this time, of course, there was no universal spelling and more importantly, no universal way of speaking one particular language (arguably this is true today as well, but that is another topic entirely). The language contact which presented itself in both grammatical and lexical (that is, nouns and verbs etc.) affected all layers of society (Again, see page 9 of Van Gelderen). Before moving on, I would like to highlight Van Gelderen's comment on why occasionally words today appear to be more like either the Old English root or their Scandinavian root: one version was, for whatever reason, more 'popular' than the other, and the other fell out of favour. Copying Van Gelderen's examples on 96: egg and sky retain Scandinavian characteristics while fish and shall retain Old English characteristics. The ins and outs of these characteristics I will leave aside for now but I can expand if there is demand for it.

Following the Battle of Hastings the higher classes no longer spoke the same language as the lower classes, as the higher classes now spoke French. As the people in charge changed, so did the language. French became the language of the courts, of nobility, and of politics. Many cultural and political terms saw their first use in England through these means. These words include those such as authority and government. I again point the reader to page 9 of Van Gelderen, it's a succinct but effective page. The vocabulary of the upper classes, then, was expanded greatly. It also drew from Latin in several ways but I'm trying to stay on topic. Again, if there is demand I could provide some details here. The increasing amount of French was not an orchestrated effort to enhance English, filling in whatever gaps English may have had. What I mean by this, is that at times terms would be introduced for things that already had names. It is relevant to note here that French speakers came to England, who would over the span of several generations have difficulty maintaining their native language, and the two thus blended together. This is not an argument I make lightly and I therefore push the reader to page 99 of Van Gelderen, where she makes this claim. Notable the French speakers: “are adding French words to the English they are acquiring” (page 99). This is where your confusion over the cow/beef and sheep/mutton divisions originate. French speakers added their words to the English language, which then slowly integrated themselves across the many layers of society. Note that this integration may take a few centuries, depending on circumstances. Incidentally, the table on page 99 (table 5.7) of Van Gelderen includes beef and mutton as examples of French that made their way into English.

Your question may seem answered at this stage, but there is another aspect of language change to consider: that of meaning shift. Meaning shift, according to Van Gelderen page 96, involves a change in meaning (duh) of one word because another, similar, word is introduced. The examples by Van Gelderen here all relate to Scandinavian roots (she is discussing Scandinavian loan words in this subsection) but a similar situation occurs with French influence. Beef and cow, (as well as sheep – mutton) originally referred to the exact same thing (cow and sheep respectively), but over time beef and mutton came to refer to parts of that animal. Recall my earlier comments about which version of a given word was preferred by speakers. There, the different layers of society all spoke (more or less) the same language and so the preferred variation would be decided by popularity. That is to say, the upper stratum did not automatically favour one version while the lower stratum preferred the other, as could be the case with beef versus cow. The difference is that in the past, the choice would be between cow and chow (this is not a real example and definitely not what happened, consult your local OED), rather than the cow and beef in the post-1066 world. Over time, camp cow and camp beef more or less integrate with one another, and then there are two words for more or less the same thing (albeit with the possible meaning shift I mentioned earlier in mind).

Unfortunately I cannot really comment on why chicken is the way it is, except that there is the more general poultry which is at least related to the French poulet. This is speculation my part, however. I currently do not have access to the OED, but I suspect you will find your answer there.

Sources used:

Van Gelderen, Elly. A History of the English Language. John Benjamins Publishing Company, Amsterdam/Philadelphia, 2006. Print.

EDIT: I had originally written this in Word with tons of words in italics, which did carry over to reddit. I did not save the document sadly, so I had to go through it by hand. I also took the time to clarify some examples and touch up the grammar.