The following sentences are correct:
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The clothes, which are dirty, won't all fit in the washing machine. (Correct use of "which" for non-restrictive clause)
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My family adopted a dog that is a Dalmatian and Retriever mix. (Correct use of "that" for a restrictive clause)
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I need shoes which are more comfortable for the marathon. (In American English, "that" is often preferred, but "which" can be acceptable in British English in this context for general use.)
The sentence "My cat, that mostly stays outside, climbs on the window when she wants to come inside." is incorrect. It should use "which" if you're providing additional information about the cat in a non-restrictive clause. Therefore, it should be: "My cat, which mostly stays outside, climbs on the window when she wants to come inside."