This is how Construct state works in Hebrew, only the first word is pluralized.
for example - Beit Sefer is "school", Batey Sefer are "schools"
You can read about it a little in the tips and notes of this class: https://www.duolingo.com/skill/he/Construct-state-1
It makes sense, since schools are houses of book - it is the building that is plural; and it is the animals that are plural, not the dear.
מחמד is like the word Hamud That means cute but we never really use this word alone.
Hayot are animals.
I was thinking that seeing as חמד means cheerful, I wondered if it could be interpreted as "animals from cheerful" perhaps 'from' (מ) a cheerful 'set' of animals.
Not really. The route of both words is identical (ח,מ,ד), and they do therefore have similar meanings, but the מ has nothing to do with from. Nice thinking, though! I love thinking of the origins of words.
מחמד is used as the "pet" adjective.
pet project - פרוייקט מחמד
It usually carries a certain criticizing, or cynical, tone. Not with חיות מחמד though - they're just pet animals.