Взимку батарея в кімнаті тепла, тому рушник швидко стає сухим.

Yes. Взимку means in winter, and узимку is just a variant with the same meaning. Ukrainian often allows both в and у for smoother pronunciation.

So these are both natural:

• Взимку батарея тепла.
• Узимку батарея тепла.

You may also hear зимою, which can also mean in winter, but взимку / узимку is a very common adverbial form.

In Ukrainian, батарея can mean both battery and radiator, depending on context.

In a sentence about a room, warmth, and a towel drying, батарея clearly means radiator or heating radiator.

So in this sentence:

• батарея в кімнаті = the radiator in the room

This is a very common everyday meaning in Ukrainian.

Because after в meaning in, when you are talking about location, Ukrainian uses the locative case.

• dictionary form: кімната = room
• locative singular: в кімнаті = in the room

So:

• в кімнаті = in the room
• у кімнаті = the same thing

If в/у means movement into something, then you usually get the accusative instead:

• в кімнату = into the room

But here the radiator is located in the room, so в кімнаті is correct.

Because it describes батарея, and батарея is a feminine noun.

Adjectives in Ukrainian must agree with the noun they describe in gender, number, and case.

• батарея = feminine singular
• so тепла = feminine singular form of теплий (warm)

Compare:

• теплий рушник = a warm towel (masculine)
• тепла батарея = a warm radiator (feminine)
• тепле ліжко = a warm bed (neuter)

Because in Ukrainian, the verb to be in the present tense is usually not expressed.

So where English says:

• The radiator in the room is warm

Ukrainian normally just says:

• Батарея в кімнаті тепла

The meaning of is is understood automatically.

This is very normal in Ukrainian. The same happens in many simple present-tense sentences:

• Він щасливий = He is happy
• Кімната велика = The room is big

Here тому means therefore, so, or that is why.

It connects the two parts of the sentence:

• батарея в кімнаті тепла = the radiator in the room is warm
• тому рушник швидко стає сухим = so the towel quickly becomes dry

So тому introduces the result.

A rough structure is:

• X, тому Y = X, so Y

Because стає means becomes or gets, not just is.

So the sentence is describing a change:

• the towel is not dry at first
• then it becomes dry

That is why Ukrainian uses стає:

• рушник стає сухим = the towel becomes dry

If you said рушник сухий, that would simply mean the towel is dry, without focusing on the process of drying.

Because after ставати / стати (to become), Ukrainian often uses the instrumental case for the adjective or noun that describes the new state.

So:

• сухий = nominative form
• сухим = instrumental singular masculine/neuter

Since рушник is masculine singular, the adjective appears as сухим.

Compare:

• Він став щасливим. = He became happy.
• Вода стала холодною. = The water became cold.
• Рушник стає сухим. = The towel becomes dry.

This instrumental after become is a very important pattern in Ukrainian.

Стає is the 3rd person singular present tense form of ставати / стати.

Here it means:

• стає = becomes / is becoming / gets

Because the subject is рушник (towel), which is singular, you need a singular verb:

• рушник стає = the towel becomes

A plural subject would give a plural form:

• рушники стають сухими = the towels become dry

Швидко means quickly. It is an adverb, and it describes how the towel becomes dry.

In this sentence:

• рушник швидко стає сухим = the towel quickly becomes dry

Its position is flexible, though some positions sound more natural than others. For example, these are all possible:

• Рушник швидко стає сухим.
• Рушник стає сухим швидко.
• Швидко рушник стає сухим. — possible, but less neutral

The version in your sentence sounds natural and neutral.

The word order is fairly flexible, but the original version is natural and clear:

• Взимку батарея в кімнаті тепла, тому рушник швидко стає сухим.

Because Ukrainian has cases, word order often changes for emphasis rather than basic grammar.

Possible variations include:

• Взимку в кімнаті батарея тепла, тому рушник швидко стає сухим.
• Рушник тому швидко стає сухим, що батарея в кімнаті тепла. — more reshaped, less basic

A learner should usually keep the original order unless there is a special reason to emphasize something.

Because Ukrainian does not have articles.

So English distinctions like:

• a radiator
• the radiator

are usually understood from context in Ukrainian.

That is why you simply get:

• батарея
• кімната
• рушник

In this sentence, context makes it natural to understand them as the radiator, the room, and the towel, but Ukrainian does not need separate words for that.