Bu yemekten hemen doydum.

Because the verb doymak usually takes the -den / -dan ending on the thing that makes you full.

• yemek = food / meal
• yemekten = from the food / from the meal

So:

• Bu yemekten doydum. = I got full from this meal.

This is a very common pattern in Turkish:

• sudan doymak = to be filled up by water
• ekmekten doymak = to get full from bread

In other words, doymak does not normally take a direct object the way English eat does.

Doydum comes from the verb doymak, which means to become full, to have had enough to eat, or more generally to be satisfied.

Here:

• doy- = verb stem
• -du- = past tense
• -m = I

So doydum means:

• I got full
• I became full
• I was filled up

In this sentence, it refers to food, so the natural English idea is I got full.

These are related, but they are not exactly the same.

• doydum = I got full / I became full
  • focuses on the change
• tokum = I am full
  • focuses on the state
• tok oldum = I became full
  • possible, but less idiomatic here than doydum

So:

• Bu yemekten hemen doydum. = I got full right away from this meal.
• Tokum. = I’m full.

In everyday Turkish, when talking about food making you full, doymak is extremely common.

Hemen usually means immediately, right away, or in some contexts very quickly.

In this sentence, hemen doydum means:

• I got full right away
• I got full very quickly

So the sentence suggests that this meal filled the speaker up fast.

In Turkish, adverbs like hemen often come before the verb or just before the part they modify.

So:

• Bu yemekten hemen doydum.

sounds natural because hemen modifies doydum:
I got full immediately / very quickly.

You may also hear other word orders, depending on emphasis, for example:

• Hemen bu yemekten doydum.
• Bu yemekten doydum hemen.

But the original sentence is a very normal, neutral word order.

No, that would not be correct.

The reason is that doymak does not normally take a direct object in the accusative (-i / -ı / -u / -ü).

So this is wrong:

• Bu yemeği doydum.

But this is correct:

• Bu yemekten doydum.

A helpful way to remember it is:

• yemek yemek = to eat food
• yemekten doymak = to get full from food

So if you want to say I ate this meal, you would say:

• Bu yemeği yedim.

If you want to say I got full from this meal, you say:

• Bu yemekten doydum.

It can mean any of those, depending on context.

yemek is a broad word and can refer to:

• food
• a meal
• a dish

So Bu yemekten hemen doydum could mean:

• I got full quickly from this meal.
• This food filled me up right away.
• I got full quickly from this dish.

The exact English translation depends on the situation.

Because bu is a demonstrative determiner here, and the case ending goes on the noun:

• bu yemek = this meal
• bu yemekten = from this meal

So the ablative ending -ten is added to yemek, not to bu.

This is normal in Turkish when bu is directly modifying a noun.

Compare:

• bu kitap = this book
• bu kitaptan = from this book

Yes, it sounds natural.

It suggests something like:

• the meal was very filling,
• the speaker became full after only a little,
• or the speaker got full faster than expected.

Depending on tone and context, it could imply:

• This meal is heavy/filling.
• I didn’t need much to feel full.
• I’m surprised I got full so quickly.

So it is not just about being full; it often carries the nuance of quick fullness.

No. Although it is very common with food, doymak can also mean to have enough of something or to be satisfied with something more generally.

For example:

• Yemekten doydum. = I got full from the food.
• Uykuya doydum. = I got enough sleep.
• Tatile doydum. = I’ve had enough vacation / I’m satisfied with the vacation.

So in your sentence, it has the food-related meaning, but the verb itself is broader.