Learn French: Le chercheur analyse le gène et la molécule. - The researcher analyzes the gene and the molecule.

You would say Il demande ce que le chercheur analyse. Because the thing being analyzed is the direct object of the sentence, you use ce que for "what" in the indirect question.

What if I wanted to say "He asks what is analyzing the gene"?

You would say Il demande ce qui analyse le gène. Here, "what" is the subject that is doing the analyzing, so you must use ce qui instead of ce que.

Why is it le gène but la molécule? Don't they both end in -e?

Yes, but noun endings in French are not a strict rule. While many nouns ending in -e are feminine, le gène is a masculine exception. Interestingly, most nouns ending in -cule (like le véhicule) are actually masculine, but la molécule is a feminine exception. It is always best to memorize the article along with the noun.

Is chercheur related to the verb chercher?

Yes! The verb chercher means to look for or to seek. Adding the -eur suffix creates the noun for the person doing the action. So, un chercheur is literally a seeker, which is the standard French term for a researcher.