Mia sorella vuole grattugiare il parmigiano, ma la grattugia è nel cassetto.

With a singular family member, Italian usually leaves out the article before the possessive:

• mia sorella
• mio fratello
• mia madre

So mia sorella is the normal form.

You usually use the article if the family noun is plural or modified:

• le mie sorelle
• la mia sorella maggiore

Italian often drops subject pronouns because the verb ending already tells you the person.

• vuole = he/she wants

So Mia sorella vuole... already clearly means My sister wants...
Adding lei is possible, but it is usually for emphasis or contrast.

After volere (to want), Italian normally uses an infinitive, just like English:

• vuole grattugiare = wants to grate
• voglio mangiare = I want to eat
• vuoi uscire = do you want to go out?

So vuole grattugiare is a very standard structure: conjugated volere + infinitive.

They are related, but they are different parts of speech:

• grattugiare = to grate → a verb
• grattugia = grater → a noun

So in the sentence:

• vuole grattugiare il parmigiano = she wants to grate the parmesan
• la grattugia è nel cassetto = the grater is in the drawer

This pair is useful because Italian often forms tool nouns from the same root as the verb.

Italian often uses the definite article where English might not. With foods and ingredients, il can sound very natural.

So:

• grattugiare il parmigiano = grate the parmesan / grate parmesan

In everyday Italian, the version with the article is very common.
Without the article, grattugiare parmigiano, it can sound more like recipe language or a more compressed style.

Yes. In everyday Italian, parmigiano often means parmesan cheese.

A useful distinction is:

• parmigiano = everyday common noun
• Parmigiano Reggiano = the full proper name of the specific protected cheese

So in this sentence, il parmigiano simply means the parmesan cheese.

Because grattugia is a feminine noun, so it takes la:

• la grattugia = the grater

Its gender is something you learn with the noun. The ending -a often suggests a feminine noun, and here that is true.

The accent distinguishes è (is) from e (and).

• è = is
• e = and

So:

• la grattugia è nel cassetto = the grater is in the drawer
• without the accent, it would be a different word

This accent is very important in writing.

Because in + il contracts to nel.

• in + il = nel
• in + lo = nello
• in + la = nella
• in + i = nei
• in + gli = negli
• in + le = nelle

So nel cassetto literally comes from in il cassetto, but modern Italian uses the contracted form.

Cassetto means drawer, like a kitchen drawer, desk drawer, or cabinet drawer.

So nel cassetto means in the drawer.
In this sentence, it most naturally refers to a kitchen drawer where the grater is kept.

Yes, Italian word order is fairly flexible, although the original version is the most neutral:

• Mia sorella vuole grattugiare il parmigiano, ma la grattugia è nel cassetto.

You could also say:

• La grattugia è nel cassetto, ma mia sorella vuole grattugiare il parmigiano.

That changes the focus a bit, but the meaning stays the same.
The original order sounds natural because it first gives the action someone wants to do, then the obstacle or relevant detail.

A rough guide is:

• grattugiare ≈ grah-too-JAH-reh
• grattugia ≈ grah-too-JAH

A few helpful points:

• tt is a doubled consonant, so it is held a little longer than a single t
• gi before a gives a soft j sound, like in jar
• the stress is on gia in grattugiare and già/già-like area in grattugia

Even if your pronunciation is not perfect at first, keeping the soft g sound is important.

Not in the same structure. After volere, you normally go straight to the infinitive:

• vuole grattugiare = correct
• vuole per grattugiare = not correct here

You would use per grattugiare when you mean in order to grate or for grating, for example:

• Usa questa grattugia per grattugiare il parmigiano.

So in your sentence, vuole grattugiare is the right choice.