Mi madre prepara una vinagreta sencilla para la remolacha y el apio.

Because mi without an accent is the possessive adjective meaning my.

• mi madre = my mother
• mí with an accent means me after a preposition:
  • para mí = for me
  • a mí = to me

So in this sentence, mi madre is correct because it shows possession.

Because the subject is mi madre, which is she.
The verb preparar is conjugated in the present tense like this:

• yo preparo = I prepare
• tú preparas = you prepare
• él/ella/usted prepara = he/she/you prepare

Since mi madre = she, you use prepara.

It is the present simple.

In Spanish, the present tense can often mean:

• a habitual action: My mother makes...
• a general fact
• something happening now, depending on context

So Mi madre prepara una vinagreta sencilla... can mean:

• My mother makes a simple vinaigrette...
• My mother is making a simple vinaigrette...
  if the context suggests it is happening right now

Because vinagreta is a feminine singular noun.

That means the article and adjective must agree with it:

• una = feminine singular a
• sencilla = feminine singular form of simple

Agreement is very important in Spanish:

• un libro interesante
• una salsa rica
• una vinagreta sencilla

So both una and sencilla match vinagreta.

In Spanish, adjectives often come after the noun.

So:

• una vinagreta sencilla
• un coche rápido
• una casa grande

This is the normal position for many descriptive adjectives.
Putting the adjective before the noun is sometimes possible, but it can change the style or emphasis. For learners, the safest pattern is:

article + noun + adjective

Here, para means for.

So:

• para la remolacha y el apio = for the beetroot and the celery

It shows the purpose or intended use of the vinaigrette: the vinaigrette is meant to go with those ingredients.

Spanish often uses definite articles more often than English does.

Here, la remolacha and el apio refer to the food items in a general or specific culinary sense:

• la remolacha = beetroot
• el apio = celery

In English, we often drop the article in this kind of sentence, but Spanish commonly keeps it.

Compare:

• Me gusta el café = I like coffee
• Corta la cebolla = Cut the onion
• para la remolacha y el apio = for beetroot and celery

Because the personal a is used mainly before specific people and sometimes pets or personified things, not ordinary food items.

Compare:

• Veo a mi madre = I see my mother
• Veo el apio = I see the celery

In your sentence, la remolacha y el apio are things, so no personal a is used.

Also, the a in this sentence is not the personal a anyway. The phrase is para la remolacha y el apio, where para means for.

Y means and.

So:

• la remolacha y el apio = the beetroot and the celery

Yes, it sometimes changes to e before words that begin with an i sound:

• padre e hijo = father and son
• España e Italia

But here the next word is el, so it stays y.

Yes. In Spain, remolacha is the normal word for beetroot / beet.

A learner should know that food vocabulary can vary across the Spanish-speaking world, but remolacha is standard and very common in Spain.

Yes, mostly you need to learn nouns together with their article:

• la remolacha = feminine
• el apio = masculine

This is a very good habit because gender affects:

• articles: el / la / un / una
• adjectives: sencillo / sencilla
• sometimes pronouns

So instead of memorizing just apio, memorize el apio.
Instead of just remolacha, memorize la remolacha.

Yes, you could, and it sounds natural.

Both verbs can work:

• prepara = prepares / makes
• hace = makes

Preparar often feels a little more directly connected to preparing food, while hacer is more general. In this sentence, both are acceptable.

The given order is the most natural neutral order:

Mi madre prepara una vinagreta sencilla para la remolacha y el apio.

Spanish word order is flexible, but changes usually add emphasis or sound more literary. For example:

• Para la remolacha y el apio, mi madre prepara una vinagreta sencilla.

This is possible, but it emphasizes for the beetroot and the celery.

For learners, the standard order is best:

subject + verb + object + extra information

A rough pronunciation guide is:

• vinagreta ≈ bee-nah-GREH-tah
• sencilla ≈ sen-THEE-yah in much of Spain

Important Spain Spanish points:

• v and b sound very similar in Spanish
• g before r in vinagreta is a hard g
• ll in most of Spain sounds like y
• c before i in sencilla is often pronounced like the th in thin

So in much of Spain, sencilla sounds like sen-THEE-yah.

Because para expresses purpose or intended use, while de usually expresses of, from, or about.

Here the meaning is:

• a simple vinaigrette for the beetroot and celery

So para is the right preposition.

If you used de, it would suggest something different, such as composition or type, not intended accompaniment.