would rather ....

Rather than and would rather

Rather than is normally used in parallel structures: for example with two adjectives, adverbs, nouns, infinitives or -ing forms. When the main clause has a to-infinitive, rather than is normally followed by an infinitive without to. An -ing form is also possible.

• I would prefer to leave now rather than wait.
• You ought to admit your crime rather than defend it.
• I would prefer to go in August rather than in July.
• I decided to write rather than phone/phoning. (NOT …than to phone)

Would rather

Would rather means 'would prefer to'. It is followed by an infinitive without to.

• Would you rather stay here or go home? (=Would you prefer to stay here or go home.)

Would rather + subject + past tense

We can use would rather to say that one person would prefer another or others to do something. We use a special structure with a past tense.

• Don’t come today, I would rather you came tomorrow. (=I would prefer you to come tomorrow.)
• I would rather you posted this letter. (= I would like you to post this letter.)

To talk about past actions, a past perfect tense is possible

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Thanks Goodman for these information.They really help me a lot understand this grammatical rule.

I have some further questions if you don't mind

Goodman

This is my take:
The use of "would rather" is not restricted by time. In certain aspects, it contains a conditional element as well.
Here is some info. Hope that helps your question.

Rather than and would rather

• I would prefer to leave now rather than wait.
• Can I change the above statement to this form, in which it gives me the same meaning:
•  I would rather leave now than wait. ( is it correct?)

We can use would rather to say that one person would prefer another or others to do something. We use a special structure with a past tense.

• Don’t come today, I would rather you came tomorrow. (=I would prefer you to come tomorrow.)
• In the above sentence, why do we use the past tense to refer to a future preference? Does the meaning change if we use a verb in the simple form?
• Such as: I'd rather you come hear tomorrow.( is it correct?)

To talk about past actions, a past perfect tense is possible

I would rather that she had stayed with me.

<EDITED by mod to credit your source.>

• I would prefer to leave now rather than wait.
• Can I change the above statement to this form, in which it gives me the same meaning:
•  I would rather leave now than wait. ( is it correct?)
  
  Yes, you can.  You can also rephrase it as "I would rather not wait"
  
• Such as: I'd rather you come hear tomorrow.( is it correct?)  
  No,  it sounds odd but "came" correct. As I mentioned earlier, this is the conditional element.
  In this usage, the correct form s past tesne of the verb.