Я почав кашляти вночі, тому випив гаряче молоко.

Because after почати / почав (to begin / began), Ukrainian normally uses an infinitive verb, just like English uses to + verb:

• почав кашляти = began to cough
• почав говорити = began to speak
• почав працювати = began to work

So кашляти is the infinitive to cough. A noun like кашель means a cough or coughing as a thing, not the action in this structure.

Кашляти is the imperfective verb, and that makes sense because coughing is viewed as an ongoing or repeated action, not a single completed event.

• почав кашляти = started coughing
• The focus is on the beginning of the process

In Ukrainian, after почати, the imperfective is very common because you are usually talking about beginning an action in progress.

Вночі is a fixed adverb meaning at night / during the night.

So:

• вночі = at night
• уночі = same meaning; this is just a spelling/phonetic variant
• в ніч usually means into the night or refers to a particular night in a different structure, so it is not the normal choice here

For this sentence, вночі is the natural adverbial form.

Because Ukrainian does not use a separate particle like English to before infinitives in this kind of construction.

Compare:

• English: I began to cough
• Ukrainian: Я почав кашляти

The infinitive is simply the basic verb form, without any extra word.

Випив is the perfective past form of випити (to drink up / to drink, completed). It shows a completed action: the person drank the hot milk as a finished act.

• випив гаряче молоко = drank the hot milk / had some hot milk
• пив гаряче молоко would focus more on the process: was drinking hot milk or drank hot milk in a more general, ongoing sense

In this sentence, the idea is: coughing started, so the speaker responded with one completed action. That is why випив fits well.

Because молоко is neuter singular, and the adjective must agree with it.

• молоко = neuter singular noun
• гаряче = neuter singular form of гарячий (hot)

So:

• masculine: гарячий чай = hot tea
• feminine: гаряча кава = hot coffee
• neuter: гаряче молоко = hot milk

This is standard adjective-noun agreement in Ukrainian.

Actually, it is in the accusative case, but for many inanimate neuter nouns, the accusative looks exactly the same as the nominative.

So:

• nominative: молоко
• accusative: молоко

Because випити / випив takes a direct object, молоко is functioning as the object of the verb, even though its form does not change.

Тому here means therefore / so / that is why. It links the two parts of the sentence logically:

• Я почав кашляти вночі = I started coughing at night
• тому випив гаряче молоко = so I drank hot milk

It is a common connector for showing result or consequence.

Do not confuse this with other meanings of тому in different contexts, such as forms related to той (that). In this sentence, it functions as a connector meaning so / therefore.

Ukrainian often omits the subject pronoun when it is already clear from the verb ending or the context.

So:

• Я почав кашляти вночі, тому випив гаряче молоко.
• More explicit: Я почав кашляти вночі, тому я випив гаряче молоко.

Both are understandable, but the version without the second я sounds more natural and less repetitive in many contexts.

Past tense verbs in Ukrainian agree with the subject in gender and number in the singular.

• я почав = I began (said by a male speaker)
• я почала = I began (said by a female speaker)
• я почало would only be used for a neuter subject, not a person in normal speech

So this sentence implies that the speaker is male. If the speaker were female, the sentence would be:

• Я почала кашляти вночі, тому випила гаряче молоко.

Yes, Ukrainian word order is fairly flexible, although some orders sound more natural than others depending on emphasis.

The given version is neutral and natural:

• Я почав кашляти вночі, тому випив гаряче молоко.

Possible variations:

• Вночі я почав кашляти, тому випив гаряче молоко.
  This emphasizes at night.
• Гаряче молоко я випив, бо почав кашляти вночі.
  This puts more emphasis on hot milk and changes the connector.

So the word order can shift, but the original sentence is a very standard way to say it.

Not directly in the same way, because бо means because, while тому means so / therefore.

Compare:

• Я почав кашляти вночі, тому випив гаряче молоко.
  I started coughing at night, so I drank hot milk.

• Я випив гаряче молоко, бо почав кашляти вночі.
  I drank hot milk because I started coughing at night.

So both are possible, but they create different sentence structures.

Yes, it is completely natural.

• гаряче молоко = hot milk
• тепле молоко = warm milk

In everyday speech, both could be used depending on what you mean. If you mean milk that is pleasantly warmed, тепле молоко may often sound more natural. If you specifically mean hot milk, then гаряче молоко is correct.