The following sentences are correct:
-
My family adopted a dog that is a Dalmatian and Retriever mix. (This sentence is correct. "That" is appropriately used for defining clauses.)
-
The clothes, which are dirty, won't all fit in the washing machine. (This sentence is also correct. "Which" is used appropriately for non-defining clauses.)
The sentence:
- I need shoes which are more comfortable for the marathon. (This sentence is less formal; "that" would be more appropriate in American English for defining clauses. So it may be considered less correct in that context, but it's often used in casual speech.)
Therefore, the correct sentences are the first and third ones, but I would suggest using "that" instead of "which" in the last one for clearer usage.