Learn French: Il me semble rationnel, mais je doute qu'il le soit. - He seems rational to me, but I doubt that he is.

In French, when you want to refer back to an adjective or a state of being previously mentioned, you use the neuter pronoun le. Here, le replaces the adjective rationnel. While in English we can just end the sentence with "I doubt that he is", French requires this pronoun to complete the thought, almost like saying "I doubt that he is it".

If we were talking about a woman, would the second half of the sentence be je doute qu'elle la soit?

No, it would remain je doute qu'elle le soit. When used as a predicate attribute to replace an adjective (like rationnelle), a verb phrase, or a whole idea, the pronoun le is neuter. This means it is completely invariable and never changes to la or les to match gender or number.

Can I leave out the le and just say je doute qu'il soit, mirroring the English "I doubt that he is"?

No, doing so would sound grammatically incomplete to a French speaker. A linking verb like être generally needs something to follow it to complete its meaning. You must use the neuter pronoun le as a placeholder so the verb isn't left hanging.

Why is the verb soit used here instead of the indicative est?