Not quite. (1) and (3) are similar in that they are generalised statements. A possible difference between them may depend on their usage. (1) can be used for counting e.g. "A dog has four legs, Two dogs have eight legs, .." (3) can't be used for counting e.g. "Dogs have four legs, Two dogs have more than eight legs, .. (X)" (2) on the other hand, is specific to the 'dog' that the speaker is referring to. In "The dog is a faithful animal", the reader feels that the speaker is referring to a particular dog that is present in the pooled knowledge between the speaker and the reader (one that is established through speaker's speech/text).

After the puzzle arose, I consulted a reference book on grammar. It said all the three sentences would be correct. The definite article "the" could be used to refer to a thing in general. Afterwards, I also found this sense in use in the Oxford Advanced Dictionary. In fact, previously, I had the same idea as Maroro's that "the" often indicated the particular thing that both speaker and reader had already known. But I still get information here. I'm grateful for all the replies here.

You can use all three. The use of "the" is rare, though.

Dogs are faithful animals. Dogs have four legs.