"Both the dog and the cat like meat."
Translation:Zowel de hond als de kat houden van vlees.
"beide" is not very commonly used in Dutch anymore. At least I'm struggling to think when I last used it. "allebei" is more current and translates as roughly the same. If you wanted to use it in this sentence it would look like: "De hond en de kat houden allebei van vlees." But neither beide nor allebei would be used at the start of a sentence, not ever. The only exception I can think of is if it's a sentence fragment where someone asks you a question and the answer would be "they both are", then you could respond with just saying "allebei/beide", but even in this example, it would be more correct to use "de beide". Mind you, it sounds truly archaic :p.
"Zowel de hond als de kat houden van vlees" comes down to "as well as" and seems to me the wrong translation. In my opinion your translation "De hond en de kat houden allebei van vlees" is the correct one if the English sentence contains "both" (allebei). Not sure if the term "alletwee" wouldn't also be correct, it might be Flemish.