Elle me conseille de changer de brosse à dents pour protéger ma gencive.

Because French object pronouns normally come before the conjugated verb.

So:

• Elle me conseille = She advises me
• literally: She to-me advises

Here, me means to me with the verb conseiller in this structure.
The full pattern is:

• conseiller à quelqu’un de faire quelque chose = to advise someone to do something

So Elle me conseille... corresponds to Elle conseille à moi..., but French does not say à moi here in normal grammar. It uses the pronoun me before the verb instead.

The pattern is:

• conseiller à quelqu’un de + infinitive

So in this sentence:

• Elle = subject
• me = indirect object, meaning to me
• conseille = advises
• de changer = to change

A useful model is:

• Je te conseille de dormir davantage. = I advise you to sleep more.
• Il nous conseille d’attendre. = He advises us to wait.

So de is required before the infinitive changer.

Because after conseiller when it is followed by an infinitive, French uses de.

So you say:

• Elle me conseille de changer...

not:

• Elle me conseille changer...

This is just the verb’s normal construction. Many French verbs work this way and require a preposition before an infinitive.

Here, changer de + noun means to switch to another one or to replace the one you currently use with a different one.

So:

• changer de brosse à dents = to change toothbrushes / switch to a different toothbrush

This use of changer de is very common when you move from one version of something to another:

• changer de voiture = change cars
• changer de place = change seats
• changer de travail = change jobs

So the idea is not really alter the toothbrush, but start using a different toothbrush.

Yes, but the nuance is slightly different.

• changer de brosse à dents emphasizes switching to another toothbrush
• changer sa / ma brosse à dents can mean replace one’s toothbrush

In real life, both can be understandable in this context.
But changer de brosse à dents is especially natural when the advice is about using a different type of toothbrush, for example a softer one.

Because brosse à dents is the fixed French expression for toothbrush.

The à here often shows the purpose or function of something:

• une brosse à dents = a brush for teeth
• une machine à laver = a washing machine
• une salle à manger = a dining room

So à does not mean to here in the usual sense. It is part of a noun pattern meaning for / used for.

Pour + infinitive expresses purpose:

• pour protéger ma gencive = in order to protect my gum

It answers the question why?

Why should I change toothbrushes?
→ Pour protéger ma gencive

This is one of the most common ways to express purpose in French when the subject stays the same.

You could also say something slightly more formal like:

• afin de protéger ma gencive

But pour is the normal everyday choice.

That is a very reasonable question.

• ma gencive suggests one specific gum area or one particular spot
• mes gencives would refer to the gums in general

In many health-related contexts, mes gencives may indeed sound more natural if the idea is general gum protection.
But ma gencive is possible if the person has one irritated or damaged area and is trying to protect that specific part.

So the singular is not wrong, but it sounds more specific.

Here it means advise.

That is because the structure is:

• conseiller à quelqu’un de faire quelque chose = to advise someone to do something

So:

• Elle me conseille de changer de brosse à dents = She advises me to change toothbrushes

But conseiller can also mean recommend, especially with a thing rather than an action:

• Elle me conseille cette brosse à dents. = She recommends this toothbrush to me.

So the meaning depends on the structure.

Ma is the feminine singular possessive adjective meaning my.

It agrees with gencive, which is a feminine singular noun:

• une gencive
• ma gencive

It does not agree with the speaker or the person’s gender. It agrees with the noun being possessed.

So even a male speaker would say:

• ma gencive

because gencive is feminine.

A rough breakdown is:

• Elle = she
• me = to me / me
• conseille = advises
• de changer = to change
• de brosse à dents = toothbrush / to a different toothbrush
• pour protéger = in order to protect
• ma gencive = my gum

So the sentence is built like this:

• She advises me to change toothbrushes in order to protect my gum

That structure is very typical French: subject + object pronoun + verb + de + infinitive + pour + infinitive