Learn Turkish: Kadın şaşkınlıktan kırılgan bardağı düşüreyazdı. - Out of astonishment, the woman almost dropped the fragile glass.

It uses the approximative compound verb structure you are currently learning. The root is düşürmek (to drop). We remove the infinitive -mek, add the connecting vowel -e, and attach yazmak (which acts as a helper verb here, unrelated to its normal meaning of "to write"). Finally, we add the past tense -dı. The formula -(y)Ayazmak means an action came dangerously close to happening, but ultimately didn't.

Is düşüreyazdı common in everyday speech, or is there another way to say "almost dropped"?

While -(y)Ayazmak is important to know for comprehension, it sounds quite literary, dramatic, or slightly old-fashioned. In everyday spoken Turkish, people usually just say neredeyse düşürdü (literally "almost dropped"). You will mostly encounter compound forms like düşüreyazdı or öleyazdı (almost died) in written stories.

Why does the verb start with düşür- instead of just düş-?

Düşmek means "to fall", which is an action the subject does themselves. To say "drop" (to cause something else to fall), you must use the causative form düşürmek. If the sentence used düşeyazdı, it would mean the woman herself almost fell down!

Why does şaşkınlık (astonishment) take the ablative -tan ending here?