Pastanın üstünde “Mutlu yıllar” yazıyordu.

Because Turkish is literally saying the cake’s top/surface. The structure is:

pasta-nın üst-ü

• pasta = cake
• -nın = genitive, meaning of the cake / the cake’s
• üstü = its top / upper surface

So pastanın üstünde means something like on the cake’s top/surface, which is the natural Turkish way to say on the cake.

Üstünde comes from:

• üst = top, upper side
• -ü = possessive ending, giving üstü = its top / its surface
• -nde = locative ending, meaning on / in / at

So üstünde literally means on its top/surface.

The n is a buffer consonant that appears before the case ending after a possessive form. That is why it is üstünde, not üstüde.

Because Turkish often uses location nouns such as üst (top), iç (inside), alt (bottom/under) where English uses prepositions.

So instead of saying something as general as in/on the cake, Turkish prefers the more specific pastanın üstünde, literally on the cake’s top/surface.

Pastada would not sound as natural here if you mean writing on the surface of the cake.

Because Mutlu yıllar is a fixed Turkish expression used for birthdays and anniversaries.

Literally, it means happy years, but idiomatically it is the normal way to say something like happy birthday or many happy returns.

So even though English uses birthday, Turkish uses years in this expression.

Formally, yes: yazıyordu comes from yazmak (to write) and is built like was writing.

But in sentences like this, it often means:

• it said
• it read
• there was written

So here it does not mean someone was actively writing on the cake at that moment. It means the cake had the words Mutlu yıllar written on it.

Because yazıyordu describes a past situation or background scene: at that moment, the words on the cake were Mutlu yıllar.

It gives the sense of that is what was written there or that is what it said.

By contrast:

• yazdı = wrote
  This would normally mean an agent performed the action of writing, so it does not fit well here.

So yazıyordu is the natural choice when describing what a written object said in the past.

There is no explicit subject, and that is normal in Turkish.

The verb is in the 3rd person singular, and the sentence is understood impersonally, something like:

• it said
• it read
• there was written

The focus is on the visible text, not on who wrote it.

They mark the exact words that were written on the cake.

Turkish often uses quotation marks for:

• direct speech
• signs
• labels
• written words being quoted exactly

So the quotation marks show that Mutlu yıllar is the literal text appearing on the cake.

Yes, you could.

Both üstünde and üzerinde can mean on it / on top of it in this kind of sentence.

Very roughly:

• üstünde is very common and feels a bit more everyday
• üzerinde can sound slightly more neutral or formal

In this sentence, both are natural.

No, Turkish word order is flexible, but this order is very natural.

Pastanın üstünde comes first to set the scene, then Mutlu yıllar gives the content of the writing, and yazıyordu comes at the end, which is typical because Turkish often puts the verb last.

So this order sounds smooth and neutral, even though other orders are possible for emphasis.