A fogfájás kezelése holnap kezdődik.

• A fogfájás kezelése = the subject noun phrase
• holnap = a time adverb meaning tomorrow
• kezdődik = the verb, begins / starts

So the structure is basically:

[subject] + [time] + [verb]

That is a very normal, neutral Hungarian sentence pattern.

Kezelése is built in two steps:

• kezel = to treat
• kezelés = treatment or treating
• kezelése = its treatment / the treatment of it

The ending -e is a 3rd person singular possessive suffix. In this sentence, the thing being treated is fogfájás (toothache), so kezelése means its treatment, that is, the treatment of the toothache.

Because Hungarian usually expresses this idea with a possessive construction, not with a separate word like English of.

So:

• fogfájás kezelése literally = toothache its-treatment
• natural English = the treatment of the toothache

This is a very common Hungarian pattern:

• a könyv címe = the title of the book
• a ház teteje = the roof of the house
• a fogfájás kezelése = the treatment of the toothache

Here fogfájás is the possessor inside a possessive structure, and in Hungarian the possessor can often appear in a bare form:

• fogfájás kezelése

There is also a longer, more explicit version:

• a fogfájásnak a kezelése

In many cases, the shorter version is more natural and compact. So the lack of an ending is normal here.

Because Hungarian usually puts one article at the front of the whole noun phrase:

• a fogfájás kezelése

English says the treatment of the toothache, where the article seems attached to treatment. Hungarian handles it as one whole unit and puts a before the entire phrase.

So do not expect something like a fogfájás a kezelése here.

Because kezd and kezdődik are used differently:

• kezd = start something
  
  • Az orvos kezdi a kezelést. = The doctor starts the treatment.
• kezdődik = begin / get started
  
  • A kezelés kezdődik. = The treatment begins.

In your sentence, the treatment is the thing that begins, so kezdődik is the natural choice.

Yes. It is a normal Hungarian compound:

• fog = tooth
• fájás = ache, pain

So:

• fogfájás = toothache

Hungarian forms many everyday nouns this way, and compounds are usually written as one word.

That is a natural neutral order in Hungarian. The sentence presents:

1. the topic/subject: a fogfájás kezelése
2. the time: holnap
3. the verb: kezdődik

Hungarian word order is flexible, though. You could also say:

• Holnap kezdődik a fogfájás kezelése.
• Holnap a fogfájás kezelése kezdődik.

These are all possible, but the emphasis shifts a little. Putting holnap first makes tomorrow more prominent.

Because kezelése is not the direct object here. It is part of the subject:

• A fogfájás kezelése = the thing that begins

If it were the direct object of another verb, then you might see accusative -t:

• Holnap kezdik a fogfájás kezelését.
  = They start the treatment of the toothache tomorrow.

Notice the difference:

• kezelése = subject form
• kezelését = accusative form

In this sentence, it is a noun.

The chain is:

• kezel = verb, to treat
• kezelés = noun, treatment
• kezelése = possessed noun, its treatment / the treatment of it

So even though it comes from a verb, here it behaves like a noun inside the subject phrase.