Which of the following is/are correct, and which expression
is more common?

1. Is cat a mammal?
2. Is a cat a mammal?
3. Is the cat a mammal ?
4. Are cats mammals?

I have difficulty deciding when an article should be
used. For instance, "cat" is a singular count
noun and is supposed to require an article. So (1)
should be wrong. But I do hear people saying
sentences like that. Are they wrong?

1 is definitely wrong. All the others are fine. This is a GENERIC question, so the cat is neither definite nor indefinite, singular nor plural - in short, you can use any combination of article and number you like ... except one.

You can't get away with "Is cat a mammal" because that usage implies that "cat" is a MASS NOUN - which is usually isn't. The construction would be fine for "Is water a liquid?" - but again, that's because water is a mass noun.

Exception - if you spell "cat" with a capital "C" it becomes a proper noun (for example the name of one of the crew of Red Dwarf), in which "Is Cat a mammal?" is absolutely fine.

Students: We have free audio pronunciation exercises.

As to the exception, you said that one can make a proper noun by capitalizing the first letter of a noun like "Cat." I am not familiar with this change-to-proper noun thing. Can you elaborate with some examples?

It only applies if it is already a proper noun. The example given was of a character in a tv show whose actual name was 'Cat', so therefore it has a capital letter. If I had a dog, I could name it Fred, Snowy, or even Dog. It would then be a dog called Dog.

A better example is the White House. It is a white house but its actual name is the White House - it gains capital letters for the name to show it is a proper noun (name)and not just a description.