У мене болить шлунок.

Because this is the normal Ukrainian pattern for talking about pain in a body part.

Ukrainian often says:

• у/в + pronoun/person = the person experiencing something
• body part in the nominative = the thing that hurts

So У мене болить шлунок is literally close to At me, the stomach hurts.

That is more natural than Мій шлунок болить in most situations.
Мій шлунок болить is grammatical, but it sounds more emphatic or contrastive, as if you are stressing my stomach specifically.

Here мене is genitive singular of я.

It follows the preposition у/в, which in this expression takes the genitive:

• у мене
• у тебе
• у нього
• у неї
• у нас

A small complication: мене has the same form in the genitive and accusative, but in у мене you should understand it as genitive.

Шлунок is in the nominative singular.

It is the grammatical subject of the sentence — the thing that is doing the hurting.

So the structure is not really I hurt my stomach. It is more like:

• The stomach hurts
• and the person affected is me

That is why шлунок stays in the nominative.

Because the verb agrees with шлунок, not with мене.

• шлунок = singular
• therefore болить = 3rd person singular

So even though the meaning in English is about I, Ukrainian grammar treats the stomach as the subject.

Compare:

• У мене болить шлунок. — My stomach hurts.
• У мене болять зуби. — My teeth hurt.

In the second sentence, the subject is plural, so the verb becomes болять.

Use болить with a singular thing that hurts:

• У мене болить шлунок.
• У мене болить голова.
• У мене болить рука.

Use болять with a plural thing that hurts:

• У мене болять зуби.
• У мене болять ноги.
• У мене болять очі.

So the verb changes according to the number of the body part, not according to the person experiencing the pain.

Yes. Ukrainian word order is fairly flexible.

The version У мене болить шлунок is a very natural, neutral order.

Other orders are possible, but they change the emphasis:

• Шлунок у мене болить. — puts more emphasis on stomach
• Болить шлунок. — natural if it is already clear whose stomach you mean, or if the speaker is just announcing the problem

So learners should treat У мене болить шлунок as the safest default.

У and в are just two pronunciation-friendly variants of the same preposition here.

They both mean the same thing. Ukrainian often switches between them for euphony — to make speech sound smoother.

So both of these are possible:

• У мене болить шлунок.
• В мене болить шлунок.

Sentence-initial у мене is extremely common, and it is a great default for learners.

Yes, very often.

There is a useful difference:

• шлунок = the stomach as an internal organ
• живіт = the belly / abdomen

In everyday speech, if someone has what English calls a stomachache, Ukrainians often say:

• У мене болить живіт.

That can sound more colloquial and more general.
У мене болить шлунок sounds more specifically about the stomach organ, and sometimes a bit more precise or medical.

So both are correct, but they are not always used in exactly the same situations.