Pilavı yapmadan önce pirinçleri iyice yıkamıştım.

-madan önce means “before doing …”. It’s built by attaching the negative converb suffix -ma/-me plus the ablative -dan/-den to the verb stem, then adding önce (“before”). Despite the name “negative converb,” it doesn’t make the clause negative here—it simply marks a temporal clause:
• Yuvarlanmadan önce pijamalarınızı giyin. – “Put on your pajamas before rolling (in the snow).”
• Pilavı yapmadan önce pirinçleri iyice yıkamıştım.

In Turkish, a direct object that is definite or specific takes the accusative suffix -ı/-i/-u/-ü. Here you’re talking about the pilav you intend to make, so you mark it pilavı. If you speak in general, no suffix is needed:
• Pilav yapmak zor değil. – “Making pilaf isn’t hard.”

-mıştı marks the past perfect, indicating an action completed before another past action. Here, “washing” was finished before “making the pilav.” The subordinate clause pilavı yapmadan önce already sets the sequence, so you could say pirinçleri iyice yıkadım, but using yıkamıştım emphasizes that the washing was fully done prior to cooking. Compare:
• Trene binmeden önce biletimi almıştım. – “I had bought my ticket before boarding the train.”

iyice is an adverb meaning “thoroughly” or “well.” It’s formed by adding the adverbial suffix -ce/-ça to the adjective root iyi (“good”). This suffix turns adjectives into manner‐adverbs:
• güzel → güzelce (“nicely”)
• yavaş → yavaşça (“slowly”)
• iyi → iyice (“thoroughly”)

Yes, Turkish word order is flexible. You could say:
• Pirinçleri iyice yıkamıştım pilavı yapmadan önce.
But the most natural order is to lead with the time clause: Pilavı yapmadan önce pirinçleri iyice yıkamıştım. The usual pattern is [Time/Condition] + [Main Clause] for clarity.