The omission of the definite article and indefinite articles

1.The man and woman are my friends.
2.The man and the woman are my friends
3.A dog and cat are fighting now.
4.A dog and a cat are fighting now.
5.The doors of the bathroom and room aren't repaired yet.
6.The doors of the bathroom and the room aren't reparied yet.

Can "the" and "a" be omitted from 2, 4, 6? And do 1, 3, 5 mean the same thing as 2, 4, 6?
I guess that we can informally omit "the" and "a" from 2,4,6 but can't grammatically.
Would you tell me whether I'm right? and if there can occur diffierences in meaning because of the omission of them, could you explain clearly how they're different in meaning?

And I just came across this sentence "But then it occured to me the room lighting and background were not symmetrical", In the sentence above, is "the room" omitted as in "(the room) background"? If so, is it grammatically correct to omit "the room" to avoid repeating?

Thank you very much for your help!