Would you rather?

I would be grateful if you corrected my sentences. I am sure that the third sentence is correct, but I am not sure about the rest.

Would you rather adopt a cat than a dog?
Would you rather adopt a cat than a dog?
Would you rather adopt a cat than a dog?
Wouldn't you rather adopt a cat than a dog?

Thanks for your corrections. So a native speaker will never say;I will be... if you correct? But why?

'Will' is too bold and forward. It carries a hint of rudeness. However, in this case 'would rather' is an idiomatic phrase.

I am sorry, but I didn't understand which phrase you meant. You were speaking about 'will' in this sentence; Will you rather adopt a cat than a dog? Am I right? Not about the will in this sentence; 'I will be grateful if you correct my sentences.'

Thanks

'Would rather' is an idiomatic phrase and 'would' is a conditional of courtesy—these are correct:

Would you rather adopt a cat than a dog?
I would be grateful if you correct my sentences.

Could you please explain it to me. I'm stumped. In one post you said, ''I would be grateful if you corrected my sentences.'''But in the last one you said, ''I would be grateful if you correct my sentences.''
What is the differences between theses two sentences?

Thank you in advance.

I would be grateful if you corrected my sentences.- more formal: adheres to Conditional II structure
I would be grateful if you correct my sentences. - informal: the politeness is retained but the conditional nature is distanced.

I believe all four sentences can be okay, but not in the same context:

1. This would be used at the humane society. You and your child go there to look at dogs, but after looking at numerous dogs, he seems to prefer a cat. So you ask him: "We came for a dog, but do you rather adopt a cat than a dog?"

2. You prefer a dog and you son has agreed to one, but you seem to detect

You've fallen off the wagon, Summit. None of those but 3 is possible. The fixed idiom is 'would rather':

Thank you so much Mr. Micawber for elaborate explanations. As usual, I truly enjoy reading your response.