i need shoes which are more comfortable for the marathon. is this sentence correct?

Yes, the sentence is grammatically correct.

How about ‘My cat, that mostly stay outside, climbs on the window when she wants to come inside.’

The sentence is almost correct. However, there is a minor grammatical error. It should be "that mostly stays outside" instead of "that mostly stay outside" because "cat" is a singular noun and requires the verb form "stays," which agrees with the subject. So the correct sentence would be: "My cat, that mostly stays outside, climbs on the window when she wants to come inside."

I meant to add a ‘s’ to stay, so sorry. Would it be correct with that included,

Yes, if you add the 's' to "stay," the sentence would be grammatically correct. So it would be: "My cat, that mostly stays outside, climbs on the window when she wants to come inside."

My family adopted a dog that is a Dalmatian and Retriever mix.

Yes, the sentence is grammatically correct.

How about ‘The clothes, wud oh are dirty, won’t all fit in the washing machine.

The sentence you provided has a few grammatical errors. A corrected version of the sentence would be: "The clothes, which are dirty, won't all fit in the washing machine."

I meant ‘The clothes, which are dirty, won’t all fit in the washing machine.’