A la meva germana li fan mal les dents, i demà anirà al dentista.

Yes, they refer to the same person, but this is normal in Catalan.

This is called clitic doubling:

• a la meva germana = the full indirect object phrase
• li = the indirect object pronoun (to her)

In Catalan, when the indirect object is stated explicitly, especially at the beginning of the sentence, the pronoun is usually still kept:

• A la meva germana li fan mal les dents.

This sounds natural. By contrast, leaving out li here would sound awkward or wrong to many speakers.

You can also say just:

• Li fan mal les dents.
  if it is already clear who li refers to.

In Catalan, with fer mal, the thing that hurts is treated as the subject.

So in:

• A la meva germana li fan mal les dents

the subject is les dents (the teeth), and that is why the verb is plural:

• fan = they do / they hurt

Literally, the structure is closer to:

• The teeth hurt her than to
• My sister hurts

This is very common in Catalan:

• Em fa mal el cap = My head hurts / My head is hurting
• Em fan mal els peus = My feet hurt

Fer mal is a very common expression meaning to hurt, to cause pain, or to be painful.

Examples:

• Em fa mal el braç = My arm hurts
• Li fan mal les cames = Her legs hurt
• Et fa mal això? = Does that hurt?

Although fer often means to do or to make, in this expression you should learn fer mal as a whole unit.

Because the subject is plural:

• les dents = the teeth

So the verb must agree with that plural subject:

• la dent fa mal = the tooth hurts
• les dents fan mal = the teeth hurt

This agreement is important:

• singular subject → fa
• plural subject → fan

With body parts, Catalan usually uses the definite article instead of a possessive, especially when the owner is already clear from a pronoun like li, em, et, etc.

So Catalan prefers:

• Li fan mal les dents

rather than:

• Li fan mal les seves dents

This is similar to other Romance languages. Once Catalan has already marked who is affected, it normally does not repeat possession.

More examples:

• Em fa mal el cap = My head hurts
• Li tremolen les mans = Her hands are shaking
• Ens rentem les mans = We wash our hands

A possessive is possible, but usually only for emphasis, contrast, or clarity.

Because la meva germana is the indirect object here, and Catalan introduces indirect objects referring to people with a.

So:

• a la meva germana = to my sister

This is completely normal with verbs and expressions that involve an affected person:

• Dono el llibre a la meva germana = I give the book to my sister
• A la meva germana li agrada el cafè = My sister likes coffee
• A la meva germana li fan mal les dents = My sister’s teeth hurt

Yes. That is also very natural.

There is a small difference in feel:

• A la meva germana li fan mal les dents
  focuses on the teeth hurting
• La meva germana té mal de dents
  means my sister has toothache

Both are correct and common.

Very roughly:

• li fan mal les dents = more tied to the body-part structure
• té mal de dents = more like describing a condition

So if you want a very idiomatic alternative, té mal de dents is excellent.

You can leave out a la meva germana if the context already makes the person clear:

• Li fan mal les dents. = Her teeth hurt.

But if you keep a la meva germana, you normally also keep li:

• A la meva germana li fan mal les dents. ✔
• A la meva germana fan mal les dents. ✘ or very unnatural

So:

• omit the full phrase if context allows
• but if the full phrase is there, the pronoun li is usually expected

Al is the contraction of:

• a + el = al

So:

• al dentista = to the dentist

This contraction is required in normal Catalan, just like:

• del = de + el

Examples:

• Va al metge = He/She goes to the doctor
• Torna del dentista = He/She comes back from the dentist

If the noun were feminine, you would not contract:

• a la dentista = to the female dentist

Anirà is the third person singular future of anar (to go).

Future forms of anar:

• aniré = I will go
• aniràs = you will go
• anirà = he/she will go
• anirem = we will go
• anireu = you all will go
• aniran = they will go

So:

• demà anirà al dentista = tomorrow she will go to the dentist

The accent in anirà shows the stress.

The word order is fairly flexible.

These are all possible:

• Demà anirà al dentista.
• Anirà al dentista demà.
• La meva germana anirà al dentista demà.

Putting demà first is very natural if you want to highlight the time:

• Demà anirà al dentista.

In the first clause, the initial placement of A la meva germana also gives it a topical, natural feel:

• A la meva germana li fan mal les dents

Catalan often moves information around for emphasis or discourse flow, while keeping the clitic pronouns.

Then the subject would be singular, so the verb would also be singular.

For example:

• A la meva germana li fa mal una dent. = One of my sister’s teeth hurts.
• A la meva germana li fa mal un queixal. = My sister has a painful molar / one molar hurts.

Compare:

• li fa mal = singular thing hurts
• li fan mal = plural things hurt

So the verb always agrees with whatever body part is doing the hurting.

It is completely standard and natural Catalan.

Both parts are very normal:

• A la meva germana li fan mal les dents
• demà anirà al dentista

This is the kind of sentence you could hear in everyday speech and also see in standard written Catalan.

A slightly different but equally natural version would be:

• La meva germana té mal de dents i demà anirà al dentista.

Both are good; they just package the idea a little differently.