Proper usage of "I wish it would....... so I ......"

I think that if you mean to stress the purpose of hanging the clothes outside (to let them get washed under the rain), using "so that" would be more appropriate:

I wish it would rain so that I could hang my dirty clothes outside

(to have them washed).

Using just "so", you would mean sooner the result you're expecting from the falling rain--to wash the dirty clothes:

I wish it would rain so I could have my dirty clothes washed

(on the clotheslines outside).

The first one would be correct except for a missing comma—although it's used in an unusual "literary" sense. (More on this later.)

There are several different common ways you could phrase it:

I wish it would rain, (and) so I will hang my dirty clothes outside [in case it does].
I had wished it would rain, (and) so I (had) hung [or hanged] my dirty clothes outside [in case it did].

The way these are phrased makes the interpretation a little difficult. They can be rephrased to make the meaning clearer:

Because I wish it would rain, I will hang my dirty clothes outside [in case it does].
Because I had wished it would rain, I (had) hung [or hanged] my dirty clothes outside [in case it did].

Back to your original examples.

You could say:

I wish it would rain so that my dirty clothes could get washed.

Without the comma, the second part of the sentence is subordinate to the first part—it explains why you wish it would rain.

But without the comma in your example:

I wish it would rain so [that] I hang my dirty clothes outside.

This doesn't make sense.

With the addition of a comma, the hanging of the clothes is no longer a reason for the wish, but an action that happens in addition to the wish.

I wish it would rain, [and] so I hang my clothes outside.

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I say that this phrasing is "literary" because writing everything in the present tense is normally only used for narrative effect:

I walk to the door and open it. I can't believe what I see, and so I jump back in surprise.

It's not wrong, but more typically it would be written in the past tense (especially outside of a literature):

I walked to the door and opened it. I couldn't believe what I saw, and so I jumped back in surprise.

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One final note. Although again not wrong, it would be more common to use hope or expect in these sentences than wish.

With wish, the the sentence could be interpreted to mean that you think that hanging your clothes outside will cause it to rain. But, in fact, you most likely mean that you just want to take advantage of the rain if it happens—and you'd like it to happen. So, hope or expect has better mental associations.