Después del gazpacho, mi abuela pidió flan y mi hermana prefirió natillas.

Because de + el contracts to del in Spanish.

So:

• después de + el gazpacho → después del gazpacho

This contraction is mandatory in normal Spanish.
A similar contraction happens with a + el = al.

Spanish often uses articles differently from English, especially with food.

In this sentence:

• después del gazpacho refers to a specific course or dish already being talked about, so the article is natural.
• pidió flan and prefirió natillas treat the desserts more generally, as the things they chose or ordered, so no article is also natural.

So all of these can be possible depending on context:

• pidió flan
• pidió un flan
• prefirió natillas
• prefirió las natillas

The version in your sentence sounds very natural for restaurant-style food choices.

Yes.

These versions change the nuance a little:

• pidió flan = she ordered flan, with the focus on the type of dessert
• pidió un flan = she ordered one flan / a flan
• prefirió natillas = she preferred custard
• prefirió las natillas = she preferred the custard / the natillas, possibly referring to a more specific option

So the original sentence is not the only possible one, just a very natural one.

Because the sentence is talking about completed actions in the past:

• your grandmother ordered flan
• your sister preferred natillas

These are seen as finished events, so Spanish uses the preterite:

• pidió
• prefirió

If you used the imperfect instead, it would sound more like a habitual action, a background description, or an unfinished past situation:

• pedía = used to order / was ordering
• prefería = used to prefer / preferred in a more ongoing sense

Because pedir and preferir are stem-changing verbs.

In the preterite, these -ir verbs change e → i in the third person forms:

• pedir → pidió, pidieron
• preferir → prefirió, prefirieron

This is a normal pattern for many stem-changing -ir verbs in the preterite.

Compare:

• pedí, pediste, pidió, pedimos, pedisteis, pidieron
• preferí, preferiste, prefirió, preferimos, preferisteis, prefirieron

Notice that the stem change happens in the third person singular and plural.

The accent marks show the correct stress.

• pidió
• prefirió

Without the written accent, the pronunciation rules would suggest a different stress pattern. The accent tells you that the stress falls on the last syllable:

• pi-DIÓ
• pre-fi-RIÓ

This is very common in third-person singular preterite forms ending in -ió.

Because natillas is normally used as a plural noun in Spanish, even when referring to one dessert serving.

This is similar to some food words in English that are naturally plural in form, even when the meaning is singular or mass-like.

So in Spain, natillas is the usual word for this dessert.
You will very often see:

• las natillas
• prefiero natillas
• tomamos natillas

Because natillas is the direct object of preferir, so no preposition is needed.

• prefirió natillas = she preferred custard

If you want to say what she preferred it to, then Spanish can use a:

• prefirió natillas al flan = she preferred custard to flan

So in your sentence, the second option is simply left understood.

In Spanish, subjects are often omitted when they are clear from the verb form.

But here there are two different people doing two different actions:

• mi abuela pidió flan
• mi hermana prefirió natillas

If you removed the subjects, it could become unclear who did what. So keeping both subjects makes the sentence clear and natural.

Because mi does not change for gender in the singular.

So you say:

• mi abuela
• mi hermana
• mi hermano
• mi padre

It only changes for number:

• mi = my, singular
• mis = my, plural

So:

• mi hermana
• mis hermanas

Yes. Spanish word order is flexible, as long as the meaning stays clear.

Your sentence:

• Después del gazpacho, mi abuela pidió flan y mi hermana prefirió natillas.

Other natural possibilities include:

• Mi abuela pidió flan después del gazpacho, y mi hermana prefirió natillas.
• Después del gazpacho, pidió flan mi abuela y prefirió natillas mi hermana.

But the original version is very natural because it introduces the time phrase first, then clearly states each subject and action.

Yes, when después is followed by a noun or pronoun, it normally takes de:

• después del gazpacho
• después de la cena
• después de eso

You can also use después on its own when the rest is understood:

• Comimos y después salimos.

So in your sentence, después del gazpacho is exactly the expected structure.