कुत्ता भूख लगी है doesn't work?
कुत्ते को भूख लगी है। would work. Keep in mind that this is still a singular sentence referring to one dog.
Yes, I forgot को, and then obviously the oblique with it.
No it doesn't, if you use it this way then the bhookh(hunger) will be changed into possessive case
"Kutte ko bhukh lagi hai"
Dog has hunger.