In my opinion, "a doctor and nurse" could mean "a doctor who also serves as a nurse", just like "a black and white cat" means "a cat which has black and white fur", am I right?
First, your examples are not similar -- "A doctor and nurse" contains two nouns, while "a black and white cat" has a single noun modified by two adjectives.
More to the point -- I don't think any native speaker would interpret either "a doctor and nurse" or "a doctor and a nurse" to mean "a doctor who also serves as a nurse." If that's what you wanted to express, you would need some clarifying word such as "both" -- "He acted as both a doctor and a nurse", "She was trained as a doctor and also as a nurse" -- and, as you see, in both of these examples I have used "a" before each noun.
I agree with Philip that "a doctor and a nurse" sounds better than "a doctor and nurse," but I wouldn't call the version with only one "a" wrong. In various possible pairs of nouns, I think if you leave out the second "a" it suggests that the two nouns form a set or are usually together or related in some way. For example, I would say
"I put on a hat and coat," but not "I put on a hat and swimsuit"
"A mother and child entered the room" but not "a mother and electrician entered the room"
"Please put a knife and fork at each place" but not "please put a knife and menu at each place"
So, whether or not I would be likely to say "a doctor and nurse" entered the room might depend on whether I thought they were working together, or just happened to be there at the same time.