"У меня собака, я не хочу кошку."
Translation:I have a dog; I do not want a cat.
whooaaaaa hold it up, stop the train cap'n. Why doesn't it say "У меня есть собака" I thought the "есть" was necessary for "to have"
You can say it with or without 'есть', here it sounds okay. But in present tense we don't need 'есть' most of the time.
"У меня собака" is would be translated more like "I own a dog." but you can also think of it as "I have a dog."
in this example you can use it. (But you cannot say: Я есть мужчина. Мы есть студенты.) I have = у меня есть... You have = у тебя есть... I had = у меня был/была/было/были... You will have = у тебя будет/будут... и так далее
Because "cat" is the object of the sentence, which means it should be in the accusative case. While "dog" is in genitive because it shows ownership/possession.