Mobilya restorasyonu bugün başlıyor.

Because mobilya restorasyonu is a Turkish noun compound meaning furniture restoration or restoration of furniture.

In this kind of compound, the second noun usually takes the ending -(s)I. So:

• restorasyon = restoration
• restorasyonu = its restoration / restoration of X

Here, mobilya tells us what kind of restoration it is, and restorasyonu is the head of the phrase.

So the structure is:

• mobilya
  • restorasyonu
• furniture
  • restoration

That is also possible, but it means something slightly different.

• mobilya restorasyonu = furniture restoration / restoration work involving furniture
• mobilyanın restorasyonu = the restoration of the furniture

The version in your sentence uses the more general compound form. It sounds like a type of work, project, or activity.
The mobilyanın version is more specific and feels like you are talking about some particular furniture.

No. Here it is not accusative.

This is a very common confusion, because -u can look like an accusative ending. But in this sentence, restorasyonu is part of the noun compound mobilya restorasyonu, which is the subject of the sentence.

A good comparison:

• Mobilya restorasyonu bugün başlıyor. = Furniture restoration starts today.
• Mobilya restorasyonunu bugün başlatıyorlar. = They are starting the furniture restoration today.

In the second sentence, restorasyonunu is accusative. In your original sentence, restorasyonu is just the compound form.

The dictionary form is başlamak, meaning to start / to begin.

The form başlıyor is built like this:

• başla- = verb stem
• -ıyor = present continuous suffix

So başlıyor means is starting / is beginning.

There is also a small sound change: when -yor is added to a verb stem ending in -a or -e, that final vowel usually drops:

• başla-
  • -ıyor → başlıyor

In Turkish, the present continuous form is used more broadly than English -ing forms.

So başlıyor can mean:

• is starting
• is beginning
• starts

With a time word like bugün (today), it often refers to something happening today or starting as scheduled today.

So bugün başlıyor can naturally be understood as:

• is starting today
• starts today

Yes, you could say Mobilya restorasyonu bugün başlayacak.

The difference is mainly nuance:

• başlıyor = more immediate, already arranged, or viewed as effectively beginning now/today
• başlayacak = simple future, will start

Both are grammatical. In many real situations, başlıyor sounds very natural for scheduled events or planned starts.

Because Turkish usually puts the verb at the end, and time expressions often come before it.

So this order is very natural:

• Mobilya restorasyonu bugün başlıyor.

But Turkish word order is flexible. You could also say:

• Bugün mobilya restorasyonu başlıyor.

Both are correct. The difference is emphasis:

• Mobilya restorasyonu bugün başlıyor. → focus first on furniture restoration
• Bugün mobilya restorasyonu başlıyor. → focus first on today

Turkish does not have articles like English the and a/an.

That means Turkish often leaves this kind of information to context.

So mobilya restorasyonu could be understood in English as:

• furniture restoration
• the furniture restoration

depending on the situation.

If Turkish wants to be more explicitly indefinite, it can use bir, but it is not always needed.

Because Turkish does not need a separate subject pronoun here.

The subject is already the noun phrase:

• Mobilya restorasyonu

So adding a pronoun would be unnecessary. Turkish often omits pronouns unless they are needed for emphasis or contrast.

In this sentence, the full subject is already present, so the sentence is complete as it is.

It can refer to any of those, depending on context.

It might mean:

• the activity of restoring furniture
• a restoration project
• the restoration work that is about to begin

That is normal in Turkish. The noun compound itself is broad, and context tells you exactly what kind of furniture restoration is meant.