Learn Spanish (Latin America): El polvo es muy espeso; por eso, uno empieza a toser. - The dust is very thick; therefore, one starts to cough.

Yes, you could, and the overall meaning would be very similar. However, using uno often makes the statement feel a bit more personal or relatable. When a speaker uses uno, they are usually including themselves in the experience. The impersonal se is a bit more detached and objective.

If a woman is speaking about this dusty situation, would she use una instead of uno?

She certainly can. While uno is traditionally used as the default gender-neutral impersonal pronoun, it is very common and completely correct for female speakers to use una to refer to people in general while emphasizing their own perspective. So a woman might naturally say por eso, una empieza a toser.

Doesn't espeso usually describe thick liquids or dense forests? Why is it used for dust here?

You are exactly right about espeso describing dense things like liquids or fog. In Spanish, airborne dust floating in a room is treated like a dense cloud or fog. You would not use grueso here, because grueso describes physical, solid thickness like a thick book or a thick wall.

Why do we need the a in empieza a toser?