Cow doesn't, but it would be पिएँगी, with the ए suffix nasalised for plural (regardless of whether it's cows or something else, drinking or some other verb).
Yes, just as you said 'the word', it's needed for the same reason. Needs to be 'the cow', 'a cow', 'that cow', 'this cow', 'my cow', or something, not just 'cow' - which cow?
The 'य्' is inserted by some speakers just as a pronunciation aid, a 'helper consonant' it's sometimes described as, just to make the double vowel less awkward.