Let's analyze each of the sentences for correctness:
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"I need shoes which are more comfortable for the marathon."
- This sentence is correct, but "that" is typically preferred in American English for essential clauses.
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"My cat, that mostly stays outside, climbs on the window when she wants to come inside."
- This sentence is incorrect. "That" should not be used in non-restrictive clauses, so it should be "My cat, who mostly stays outside, climbs on the window when she wants to come inside."
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"My family adopted a dog that is a Dalmatian and Retriever mix."
- This sentence is correct. "That" is appropriately used in a restrictive clause.
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"The clothes, which are dirty, won't all fit in the washing machine."
- This sentence is correct. "Which" is properly used for non-restrictive clauses.
Summary
- Correct: 1, 3, and 4.
- Incorrect: 2.
If you are only looking for correct sentences, sentences 1, 3, and 4 are correct.