Let's analyze the sentences for correctness:
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My family adopted a dog that is a Dalmatian and Retriever mix.
- This sentence is correct. The use of "that" is appropriate here for restrictive clauses.
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The clothes, which are dirty, won't all fit in the washing machine.
- This sentence is correct. The use of "which" in a non-restrictive clause is appropriate and the commas are correctly placed.
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I need shoes which are more comfortable for the marathon.
- This sentence is technically acceptable, but in American English, it is more common to use "that" for restrictive clauses. In British English, "which" can be used in this context. Therefore, it can be seen as acceptable, but "I need shoes that are more comfortable for the marathon" would be preferred in American English.
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My cat, that mostly stays outside, climbs on the window when she wants to come inside.
- This sentence is incorrect. The correct word should be "which" in a non-restrictive clause. It should read: "My cat, which mostly stays outside, climbs on the window when she wants to come inside."
Summarizing the correctness:
- Sentences 1 and 2 are correct.
- Sentence 3 is acceptable but debatable based on dialect.
- Sentence 4 is incorrect.
So, the correct sentences are 1 and 2.