Should we say
Dogs have noses.
or
Dogs have a nose.
or
Dogs have nose.
like wise, at school where each student has a backpack, can I say
Students, carry your backpacks
or
Students, carry your backpack.
Or if I say
Everyone, pick up your backpacks.
Or
Everyone, pick up your backpack.
That's a problem in English. "Xs have Y" and "Xs have Ys" can both be ambiguous.
We can quickly dismiss one option you offer: "Dogs have nose" is just wrong. A singular noun requires an article, a possessive, or one of a small set of adjectives, like the number "one". The rest of your sentences are all grammatically correct, but ambiguous.
In the case of dogs and noses, of course we all know that each dog has one nose.
But suppose I said, "Students at our school have a computer." Does that mean that each student has a computer, or that all the students share one computer? It's no clearer if I say, "Students at our school have computers." This could mean that each student has one computer, or that each student has several computers. Or it could mean that there is a pool of computers that the students share.
To my mind, if you say "Xs (plural) have a Y (singular)", I'd generally take that to mean that all the Xs share one Y. I think that's the reading that makes logical sense. But many English speakers, rightly or wrongly, don't say it that way.
Likewise, I'd say, "Students, carry your backpacks." If you said, "Students, carry your backpack", that would sound to me like there was one giant backpack that was so heavy that it took all the students working together to carry it. Of course I know that's unlikely, so if I heard the sentence with a singular "backpack", I'd like say to myself, "Oh, he means backpacks". But what if it was something that could be huge? Like, "Students, carry your science project." The students could all be sharing one large science project that requires many people to carry.
If you want to be clear, use a phrase like "each X" or "every X". Like, "Each dog has a nose" or "Each dog has one nose." Or, "Students, you must each carry your own backpack". Etc.
All three are idiomatic statements of a general truth:
A dog has a nose.
Dogs have a nose.
Dogs have noses.
The last one is potentially ambiguous in situations where the subject is not something everyone knows about already, but is perhaps a newly discovered creature or something like that.
Orthodontosauruses had horns.
Did the creature have one horn or two?
