Singular or plural usage for ellipsis in direct object etc

Suppose I have the following sentences:

1. There should be an X and a Y chromosome.
2. There should be an X and a Y chromosomes.

Is the second grammatically correct? If the last word had to be plural for the same meaning of the sentence and an ellipsis, would the following be correct?

3. There should be X and Y chromosomes.

The first and the last sentence are correct. The middle sentence is not correct.

The reason is this sentence, which is the original one:

• There should be an X chromosome and a Y chromosome.

Notice that this is not

• *There should be an X chromosomes and a Y chromosomes.

Neither chromosome should be plural.

That's what is meant.
Now the rule of Conjunction Reduction deletes the first chromosome, leaving sentence 1

• There should be an X [...] and a Y chromosome.

Since chromosome wasn't plural before Conjunction Reduction, it isn't plural afterwards.
So the second sentence above is ungrammatical.
Conjunction Reduction only deletes; it doesn't do arithmetic.

However, the speaker can do the arithmetic.
There is, after all, one X chromosome and one Y chromosome involved,
and that makes two chromosomes, should one need to speak of them.
(There is, of course, no article, since a/an is only singular.)

But X and Y is a perfectly reasonable conjoined NP that can modify
a plural chromosomes, which leads to the third sentence:

• There should be X and Y chromosomes.

No, your second sentence is grammatically incorrect because of the ellipsis. The entire sentence is:

There should be an X chromosome and there should be a Y chromosome.

To combine them with the coordinating conjunction and, all you need to do is delete the second instance of there should be:

There should be an X and a Y chromosome.

Consider:

There should be a boy and a girl child. [grammatically correct]
There should be a boy and a girl children. [grammatically incorrect]

These are sentences with similar structures.

The third sentence doesn't mean the same as the first. The first sentence says that there are only two chromosomes, one X and one Y. The third sentence says that there should be at least two X chromosomes and at least two Y chromosomes, possibly more.