Suppose that there are two dogs. One is white and the other is black. Which one is grammatically correct?
the white and the black dogs
or
the white and the black dog
Should I use singular dog or plural dogs?
Both structures are grammatically correct, but the former is what you want, I think.
"The white and the black dog" is correct because here, ellipsis just dropped the "dog" after "white". "The white dog and the black dog" is the full or "original" phrase, and it's definitely talking about two (and only two) distinct dogs.
"The white and the black dogs" is grammatical but messy. Maybe it can be parsed as [[the white and the black]AdjP dogs]NP (where "the white and the black" is one chunk modifying "dogs"). That still doesn't say how many dogs there are. Maybe one of each colour, but maybe more. More confidently, I can say it can be read, "the white dogs and the black dogs" (because of ellipsis like above) where we have a minimum of 4 dogs (2 or more white, 2 or more black), or even "the white (dog) and the black dogs" Where we have at least 3.
"The white and the black dog.."
Reason : Although there are 2 dogs here, they have different properties.
Your statement seems to be incomplete, it should start with something say for eg. :
There were two dogs among them, the white dog and the black dog, playing together with the ball.
"the white and the black dog" is correct. To make the other correct you could add a comma "the white, and the black dogs" to signify that the dog is not white and black.
If there are two dogs, one of which is entirely white and the other of which is entirely black, the only way to be completely clear is to repeat the noun:
the white dog and the black dog
To me, this doesn't even make sense:
the white and the black dog
Although ellipsis is possible if you put the noun after the first adjective (although I still think it's usually preferable to repeat the noun):
the white dog and the black
This means one dog that has both black and white coloring:
the white and black dog
This means many dogs that have black and white coloring, some of which may be entirely white or entirely black:
the white and black dogs