Learn Spanish (Latin America): Esa repisa la limpié a pesar de que estaba muy polvorienta. - That shelf, I cleaned even though it was very dusty.

This is a grammar structure called topicalization. When you move a direct object like Esa repisa to the very beginning of the sentence to emphasize it, Spanish requires you to double it by placing the matching direct object pronoun (la) right before the verb.

Could I just use standard word order and say Limpié esa repisa without the la?

Yes, absolutely. The standard word order is Limpié esa repisa a pesar de que estaba muy polvorienta, without any extra pronoun. You only add the la when you pull the direct object to the front for emphasis, as in this specific sentence.

Why is estaba in the indicative mood instead of the subjunctive after a pesar de que?

The phrase a pesar de que (even though) takes the indicative when you are declaring a known, established fact. The speaker knows for a fact that the shelf was dusty. You would only use the past subjunctive (estuviera) if the dustiness was just a hypothetical situation, or if you were acknowledging someone else's opinion rather than stating your own fact.

Why do we use the preterite limpié but the imperfect estaba in the same sentence?