The correct sentences from your list are:
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The clothes, which are dirty, won't all fit in the washing machine. (This is correct; the clause "which are dirty" is a non-restrictive relative clause, and the commas are used correctly.)
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My family adopted a dog that is a Dalmatian and Retriever mix. (This is also correct; "that" is appropriate here for a defining relative clause.)
The following sentences contain errors:
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My cat, that mostly stays outside, climbs on the window when she wants to come inside. (This should use "who" or "which" instead of "that" because it is a non-restrictive clause about the cat.)
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I need shoes which are more comfortable for the marathon. (In American English, "that" is preferred for defining clauses, so this would ideally be: "I need shoes that are more comfortable for the marathon.")
So, the first and fourth sentences are correctly constructed.