あ、犬飼ってるんですか？何犬飼ってるんですか？英語でなんて言うの？

Oh!' is used to express surprise or pleasure.

Repeating the statement also implies surprise or pleasure: 'You have a dog!'

Breed/kind/type of dog all have the same meaning.

A 'breed' of dog is the type or kind of dog. For example: "My dog is a Corgi."

2nd and 3rd expression are for dog type.

I love dogs too. What breed is it?
This means that you have discovered that they love dogs and yo would like to talk about dogs.
A breed means what type of dog it is.
I also have a dog.Mine is a greyhound. How about yours?
This is also a fun and casual way of starting the conversation.

You : Do you have a dog?
Person : Yes, I do.
You : What is the breed of your dog?
Person : It is a sharpei.

breed : type, family, line

sharpei : a type of dog

Hello :-) I hope this helps!

A)Dogs are my favorite! What breed is your dog?
Definitions
Favorite -best-loved, most-liked
Example- I am not really a cat lover, dogs are my favorite.
Breed- produce offspring, line, family; type, kind, sort, class
Example-"a medium-sized breed of cow"

B)What kind of dog do you have? I have a pit-bull.
Definitions
*Kind- Type,sort,
Example - What kind of perfume do you use ?

-> In conclusion all of the above mentioned sentences are different ways you can ask some what type of dog they have. I have added additional sentences which do not necessary fit to the topic however to help you understand the words i have used :-) I hope it helps !

Wow! - this is to express surprise

Aww! - this is to express that you find something adorable/cute

breed - this is the word you'd use to describe a type of dog. You can also say 'type' but 'breed' is the correct word.

A good way to react after learning that a person has a dog would be:
"You have a dog?"
"Oh, I love dogs!"
You could then follow up with:
"What breed is it?"
"What kind of dog is it?"