I. It's raining cats and dogs - Origin

Ref. The Guardian It is thought that the phrase originated in England during the seventeenth century. During heavy rains, floods washed dead cats and dogs across the streets. https://www.theguardian.com/notesandqueries/query/0,5753,-22408,00.html#:~:text=The%20phrase%20is%20supposed%20to,ancient%20associations%20with%20bad%20weather.

II. Dolphins and Porpoises = Cats and Dogs

If we examine porpoises and dolphins at this level, they are as physically different as dogs and cats. What does dogs and cats mean in this other sentence?

The world is divided into two groups: cat lovers and dog lovers. Both creatures have completely distinct traits, thus the author of the above phrase utilised this to demonstrate the dissimilarity between a dolphin and a porpoise. You'll be able to grasp this after reading the sayings in the next topic.

III. Other proverbs on Cats and Dogs.

Is there any other usage of "dogs and cats"?

I would be surprised if there weren't any.
Ref. pethelpful.com.
1.Dogs remember faces, cats' places..
An English proverb. It perfectly describes the difference between the canine and the feline mentality.
2.The dog for the man, the cat for the woman.
This English proverb that says more about people than it does about the dog and the cat.
4.The dog may be wonderful prose, but only the cat is poetry..
This French proverb is very elegant in its description.
5. In the long run, even a dog will compromise with the cat..
This Hungarian proverb recognizes the superior creature.
6. Dogs believe they are human. Cats believe they are God..
People like to humanize their dogs. It seems like the cat has higher aspirations.
7.Dogs have owners; cats have staff.
No point in arguing. We all know it. We own our dogs while our cats own us.
8. Having done wrong, the cat will always try to make it look like the dog did it..
Sneaky cats know how to do bad things and make you think someone else did it.
***9.Buy a dog a toy, and it will play with it forever. Buy a cat a present, and it will play with the wrapper for 10 minutes.
A true statement. Try it for yourself. A cat will ignore the toy but will play with the box it came in.
10.Every dog has its day—but the nights are reserved for the cats..
The night and darkness are synonymous with cats.
https://pethelpful.com/cats/77-Cat-Proverbs-And-Sayings#:~:text=44.,the%20dog%20and%20the%20cat.

I. It's raining cats and dogs - Origin

Ref. The Guardian 
It is thought that the phrase originated in England during the seventeenth century. During heavy rains, floods washed dead stray cats and dogs across the streets. https://www.theguardian.com/notesandqueries/query/0,5753,-22408,00.html#:~:text=The%20phrase%20is%20supposed%20to,ancient%20associations%20with%20bad%20weather.

II. Dolphins and Porpoises = Cats and Dogs

If we examine porpoises and dolphins at this level, they are as physically different as dogs and cats. What does dogs and cats mean in this other sentence?

The world is divided into two groups: cat lovers and dog lovers. Both creatures have completely distinct traits, thus the author of the above phrase utilised this to demonstrate the dissimilarity between a dolphin and a porpoise. You'll be able to grasp this after reading the sayings in the next topic.

III. Other proverbs on Cats and Dogs.

Is there any other usage of "dogs and cats"?

I would be surprised if there weren't any.
Ref. pethelpful.com.
1.Dogs remember faces, cats' places..
An English proverb. It perfectly describes the difference between the canine and the feline mentality.
2.The dog for the man, the cat for the woman.
This English proverb that says more about people than it does about the dog and the cat.
4.The dog may be wonderful prose, but only the cat is poetry..
This French proverb is very elegant in its description.
5. In the long run, even a dog will compromise with the cat..
This Hungarian proverb recognizes the superior creature.
6. Dogs believe they are human. Cats believe they are God..
People like to humanize their dogs. It seems like the cat has higher aspirations.
7.Dogs have owners; cats have staff.
No point in arguing. We all know it. We own our dogs while our cats own us.
8. Having done wrong, the cat will always try to make it look like the dog did it..
Sneaky cats know how to do bad things and make you think someone else did it.
9.Buy a dog a toy, and it will play with it forever. Buy a cat a present, and it will play with the wrapper for 10 minutes..
A true statement. Try it for yourself. A cat will ignore the toy but will play with the box it came in.
10.Every dog has its day—but the nights are reserved for the cats..
The night and darkness are synonymous with cats.
https://pethelpful.com/cats/77-Cat-Proverbs-And-Sayings#:~:text=44.,the%20dog%20and%20the%20cat.