Je vais me faire sécher les cheveux chez la coiffeuse avant la réunion.

Yes, vais literally means go, but aller + infinitive is a very common way to talk about the near future in French.

So:

• Je vais me faire sécher les cheveux = I’m going to get my hair dried

This is similar to English I’m going to...

You could also use a simple future form in some contexts, but aller + infinitive often sounds more natural for a planned or imminent action.

Se faire + infinitive often means to have something done to yourself or to get something done.

So:

• me faire sécher les cheveux literally = to make/do for myself the drying of the hair
• natural English = to get my hair dried or to have my hair dried

The important idea is that you are not doing the action yourself. Someone else is doing it for you.

Compare:

• Je sèche mes cheveux = I dry my hair
• Je me fais sécher les cheveux = I get my hair dried

Because sécher here is being used transitively, meaning to dry something.

• sécher les cheveux = to dry the hair

You do not need se with sécher in this structure.

French uses:

• Je me fais couper les cheveux = I get my hair cut
• Je me fais laver la voiture = I get my car washed
• Je me fais sécher les cheveux = I get my hair dried

The me belongs to faire (I get/have done for myself), not to sécher.

French often uses the definite article (le, la, les) where English uses a possessive (my, your, his...), especially with body parts when the owner is already clear.

So French prefers:

• Je me lave les mains = I wash my hands
• Elle se brosse les dents = She brushes her teeth
• Je vais me faire sécher les cheveux = I’m going to get my hair dried

Because me already shows whose hair it is, les cheveux is the normal choice.

Using mes cheveux is not impossible in all contexts, but here it would sound less natural.

In French, hair on your head is usually expressed as les cheveux, which is plural.

So:

• un cheveu = a single hair
• les cheveux = hair / the hair on one’s head

English uses the singular mass noun hair, but French usually uses the plural for this meaning.

Chez often means at the home/business/place of someone.

So:

• chez la coiffeuse = at the hairdresser’s / at the hairdresser’s salon

It does not literally mean the hairdresser is in her house. It can refer to her professional place too.

Other examples:

• chez le médecin = at the doctor’s
• chez le boulanger = at the baker’s
• chez moi = at my place / at home

Yes. The word changes depending on the gender of the hairdresser.

• le coiffeur = male hairdresser
• la coiffeuse = female hairdresser

So if you are going to a woman, chez la coiffeuse is correct.
If you are going to a man, you would say chez le coiffeur.

In everyday French, people may also refer to the salon itself:

• Je vais chez le coiffeur / chez la coiffeuse
• Je vais au salon de coiffure

In French, object pronouns usually come before the conjugated verb.

Here the conjugated verb is vais, and the infinitive phrase is faire sécher. In a structure like this, the pronoun goes before the infinitive it belongs to:

• Je vais me faire sécher les cheveux

This is the normal placement with aller + infinitive.

Compare:

• Je vais me coucher = I’m going to go to bed
• Je vais me faire couper les cheveux = I’m going to get my hair cut

Here it simply means before the meeting.

• avant = before
• la réunion = the meeting

So the sentence gives a time reference: the speaker plans to get their hair dried before the meeting starts.

A useful contrast:

• avant la réunion = before the meeting
• après la réunion = after the meeting

Yes, but it means something different.

• Je vais faire sécher mes cheveux usually means I’m going to dry my hair or I’m going to let my hair dry / make my hair dry, depending on context.
• Je vais me faire sécher les cheveux means I’m going to get my hair dried by someone else.

So the version with me faire clearly suggests a service done for you, which fits with chez la coiffeuse.

Yes, very common. French often uses se faire + infinitive for services people receive.

Examples:

• se faire couper les cheveux = to get one’s hair cut
• se faire coiffer = to get one’s hair done
• se faire maquiller = to get one’s makeup done
• se faire masser = to get a massage / to get massaged

So Je vais me faire sécher les cheveux sounds very natural for going to a salon and having someone dry your hair.