Learn Portuguese (Portugal): O gato dorme por baixo da mesa.

Both debaixo de and por baixo de mean "under" or "underneath", and in most everyday situations, you can use them interchangeably. The phrase por baixo de can sometimes imply being covered by something, but here, saying debaixo da mesa would be perfectly correct and mean exactly the same thing.

Why is it da mesa instead of just de mesa?

The phrase for "under" is por baixo de. Because we are talking about a specific table ("the table"), we need the feminine definite article a. In Portuguese, de and a must combine to form da. If we just said por baixo de mesa, it would sound like "underneath table" instead of "underneath the table".

I sometimes see the word sob used for "under". Could I say O gato dorme sob a mesa?

Grammatically, yes! However, sob is mostly used in formal writing, the news, or fixed expressions. For everyday spoken Portuguese, people almost always use por baixo de or debaixo de when talking about a physical location like a cat under a table.

Is the verb dorme completely regular here?