Батарея снова работает, и благодаря ей в комнате тепло.

Here it means radiator.

In Russian, батарея can mean:

• a battery
• a radiator
• sometimes even a battery/unit of artillery in other contexts

In a sentence about a room being warm, the natural meaning is radiator. So Батарея снова работает means something like The radiator is working again.

Because Russian often uses работать for anything that is functioning / operating properly.

So:

• телефон работает = the phone works
• интернет работает = the internet works
• отопление работает = the heating works
• батарея работает = the radiator is working / functioning

With a radiator, this means it is functioning as it should and giving heat.

Снова means again.

In Батарея снова работает, it modifies работает:

• Батарея снова работает = The radiator is working again

Its position is natural and neutral here. Russian word order is flexible, but this placement is very common when again refers to the action/state of the verb.

Other orders are possible for emphasis, for example:

• Снова батарея работает — marked, unusual in normal speech
• Снова работает батарея — possible, with more focus on again

The given version is the most straightforward.

Because благодаря governs the dative case.

You can think of it as meaning thanks to or owing to, and it answers:

• благодаря кому?
• благодаря чему?

Since батарея is feminine, the pronoun она becomes ей in the dative:

• она → ей

So:

• благодаря батарее = thanks to the radiator
• благодаря ей = thanks to it

Two different reasons:

1. Not её

   • её is usually genitive or accusative
   • after благодаря, you need the dative
   • so it must be ей

2. Not ней

   • after many prepositions, 3rd-person pronouns get an extra н-:
     • у неё
     • с ней
     • в ней
   • but after certain prepositions such as благодаря, согласно, вопреки, Russian normally uses the form without н-

So the correct phrase is:

• благодаря ей

Because в with location normally takes the prepositional case.

Here the meaning is in the room, so:

• комната → nominative
• в комнате → prepositional, showing location

Compare:

• в комнате тепло = it is warm in the room
• в комнату = into the room (motion, accusative)

So в комнате is used because the sentence describes where the warmth is.

Because тепло here does not describe the room as an adjective. It expresses a state/condition: it is warm.

So:

• в комнате тепло = it is warm in the room
• комната тёплая = the room is warm / the room is a warm room

The difference is subtle but important:

• тепло focuses on the current condition
• тёплая describes the room itself with an adjective

In weather, temperature, and general conditions, Russian very often uses words like:

• холодно = it is cold
• жарко = it is hot
• тепло = it is warm
• темно = it is dark

Because in the present tense, Russian usually omits the verb to be.

So where English says:

• The room is warm
• It is warm in the room

Russian often simply says:

• Комната тёплая
• В комнате тепло

There is no present-tense есть here. Using есть would sound wrong in this sentence.

So в комнате тепло is a complete sentence in Russian even without a separate verb meaning is.

Because this sentence joins two clauses:

1. Батарея снова работает
2. благодаря ей в комнате тепло

Each part has its own predicate:

• работает
• тепло (which functions as the predicate of the second clause)

So Russian punctuation treats this as a compound sentence, and the comma before и is standard:

• Батарея снова работает, и благодаря ей в комнате тепло.

Usually, yes: it often suggests a positive cause or at least a cause viewed favorably.

That is why it fits well here:

• благодаря ей в комнате тепло = thanks to it, the room is warm

If the result is negative, Russian more often uses из-за:

• Из-за батареи в комнате слишком жарко = Because of the radiator, the room is too hot

That said, in modern usage благодаря can sometimes sound fairly neutral, but the positive flavor is still important and worth remembering.