После укуса у меня начался зуд на руке.

Because после requires the genitive case.

• Dictionary form: укус = bite
• Genitive singular: укуса

So:

• после укуса = after the bite

This is a very common pattern:

• после урока = after the lesson
• после дождя = after the rain
• после работы = after work

Russian often uses у + genitive to express something happening to a person, especially with states, sensations, and possession.

So:

• у меня начался зуд literally means something like at me, itching began
• natural English translation: I started to feel itching / I developed an itch

This pattern is very common:

• У меня болит голова. = I have a headache.
• У меня началась температура. = I came down with a fever.
• У меня появился кашель. = I developed a cough.

Using я would not fit this structure.

Because the subject of the sentence is зуд.

• зуд = itch / itching
• It is masculine singular
• So the past-tense verb must also be masculine singular: начался

Compare:

• начался зуд = the itching began
• началась боль = the pain began
• началось воспаление = the inflammation began

The verb agrees with the noun that is doing the grammatical action, not with меня.

Зуд is a noun meaning itch, itching, or an itching sensation.

So:

• начался зуд = itching started / an itch began

Related words:

• зудеть = to itch
• чесаться = to itch, to feel itchy, or to scratch oneself depending on context

Examples:

• Кожа зудит. = The skin itches.
• У меня зуд. = I have itching / I have an itch.
• Рука чешется. = My hand/arm itches.

Because на here means on the arm/hand.

• на руке = on the hand / on the arm

This uses the prepositional case after на when describing location:

• на столе = on the table
• на лице = on the face
• на ноге = on the leg

So зуд на руке means itching on the arm/hand.

It can mean either, depending on context.

In Russian, рука covers a wider range than English hand. It can refer to the whole upper limb, while English usually distinguishes hand and arm.

So на руке could mean:

• on my hand
• on my arm

Context would tell you which one is meant.

Because the itching is understood as being on the surface of the skin, not inside the arm/hand.

• на руке = on the arm/hand
• в руке = in the hand / inside the hand

For skin symptoms, Russian usually uses на:

• сыпь на руке = a rash on the arm/hand
• рана на ноге = a wound on the leg
• зуд на коже = itching on the skin

If you said в руке, it would suggest something more internal.

Yes, it is understandable and natural enough. It means that after a bite, an itching sensation began on the arm/hand.

That said, depending on context, Russians might also say:

• После укуса у меня зачесалась рука. = After the bite, my hand/arm started itching.
• После укуса на руке начался зуд. = After the bite, itching started on my arm/hand.

Your sentence is fine, but some speakers might prefer one of these alternatives because they sound a bit more direct or idiomatic.

They are similar, but not identical.

• начался зуд focuses on the symptom itself: itching began
• зачесалась рука focuses on the body part: the arm/hand started itching

So:

• После укуса у меня начался зуд на руке. = After the bite, itching began on my arm/hand.
• После укуса у меня зачесалась рука. = After the bite, my arm/hand started itching.

Both are good Russian. The second one is often more conversational.

No, because зуд is masculine, not feminine.

Correct forms:

• зуд → начался
• боль → началась
• жжение → началось

So you must say:

• начался зуд

not

• началась зуд

Укус means bite in general. In context, it often refers to an insect bite, but it does not have to.

Examples:

• укус комара = mosquito bite
• укус собаки = dog bite
• укус клеща = tick bite

If you specifically mean a sting, Russian may use another word depending on the animal/insect, such as ужаление. But in everyday speech, укус is very common for many bite-like situations.

Yes. Russian word order is flexible, and changing it often changes the focus rather than the core meaning.

For example:

• После укуса у меня начался зуд на руке.
• У меня после укуса начался зуд на руке.
• На руке после укуса у меня начался зуд.

All of these are possible, though some sound more natural than others depending on context.

The original sentence is fine because it starts with the time relationship после укуса and then introduces the experiencer у меня.

Because Russian often does not use a possessive adjective when the owner is already obvious.

Here, у меня already tells us whose arm/hand it is. So на руке is enough.

Russian often avoids repeating possession when it is clear:

• Я помыл руки. = I washed my hands.
• У меня болит голова. = My head hurts.
• Он закрыл глаза. = He closed his eyes.

You could sometimes add моей for emphasis, but normally it is unnecessary here.