Plural or singular noun when it refers to two things in a list

Solution 1:

"A" is singular: You can't use "cars" and "arms." "A red car and a blue car" and "a left arm and a right arm" indicate two cars and two arms. "Red and blue cars" and "left and right arms" have indeterminate numbers.

Solution 2:

There are two problems here; one is structural and the other is semantic. The reason why it's wrong to say "I have a red and a blue cars" is structural; the sentence has an unwritten but understood instance of the word "car" after the word "red". This sentence asserts that you have two cars, one red and one blue. The example "I have old and new cars" (when you have a total of two of them) is more problematic. The meaning is, at best, ambiguous, since it could certainly mean that you have two or more of either. A careful writer might want to avoid ambiguity. The sentence "Everyone has left and right arms" is similarly flawed, but appeals more to the native ear, due to the fuzziness of the word "everyone", which conveys a collective meaning, even though it literally means "every one" and is indisputably singular. It also gets a pass because it states the obvious, so the ambiguity isn't taken seriously.

Solution 3:

If there is a single noun phrase referring to several things, it's plural. If there are several noun phrases, each referring to one thing, each of the noun phrases is singular. So, it's all about noun phrases, and how many of them there are.

An article starts a noun phrase, so in straightforward cases, you can count the articles to tell how many noun phrases there are. (Some noun phrases don't have articles.) In "Everyone has a left and a right arm" there are two articles in the object, so there are two noun phrases coordinated: [[a left (arm)] and [a right arm]]. In "The left and right arms are both necessary" there is just one article in the subject noun phrase, so there is just a single plural noun phrase, whose noun head is modified by a coordination of two adjectives: [the [left and right] arms], and this is why that noun phrase is plural and its head noun "arms" must be plural.

In ordinary cases of coordination, two things of the same category are connected by "and" to form a new phrase of that same category. That's why when two noun phrases, each with its own article, are connected with "and", the result is a noun phrase, and why when two adjectives are connected with "and", the result is an adjective. People with training in traditional grammar may resist calling a multi-word constituent like "left and right" an adjective, but they ought to get used to it.