Learn English – either or both A and B vs. either or both A or B

"It can live both in water and land."

A good trick I use for these types of situations is to imagine it says: "It can live both in water and in land" since that's basically what you're saying. For this you need to add "and on land." Otherwise you're saying it can live in land as well.

"He is responsible for the loss and damage to the article."

For this sentence, use the same trick I said for the previous sentence: "He is responsible for the loss to the article and damage to the article." The phrase "loss to the article" doesn't seem to have the meaning you were going for, so again I'd say add "loss of."

Your first and third phrasings are incorrect. To say "never better than either" means "always worse than both" (or, at best, equal to one of them).

#2 and #4 are both accurate, and logically equivalent, but #4 is convoluted—harder for the reader to parse.

So I would say use #2.

• A can never be better than both B and C.

The "both" makes it explicit. Virtually all readers will take it to mean "A cannot simultaneously be better than B and better than C". Which is what you meant. (Actually, your introductory sentence is "clearer than both #2 and #4" but #2 is shortest.)