После обеда стало ещё пасмурнее, и начался ливень.

Because после always requires the genitive.

• после обеда = after lunch / after dinner / after the meal
• Base form: обед
• Genitive singular: обеда

This is a very common pattern:

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

So in this sentence, после обеда is simply the normal way to say after lunch.

Here стало is part of an impersonal construction.

Russian often says the equivalent of it became cloudier, but without an actual word for it. English needs a dummy subject (it), but Russian usually does not.

So:

• стало пасмурнее = it became cloudier
• literally, something like became cloudier

The verb is in the neuter singular past form (стало) because impersonal past-tense predicates often use neuter singular.

This is very common with weather and general conditions:

• стемнело = it got dark
• похолодало = it got colder
• потеплело = it got warmer

Пасмурнее is the comparative form of пасмурный / пасмурно, meaning more gloomy / more overcast / cloudier.

So:

• пасмурно = gloomy, overcast
• пасмурнее = gloomier, more overcast, cloudier

In this sentence, стало ещё пасмурнее means it became even more overcast or it got even cloudier.

Russian often forms comparatives in this short, single-word way:

• быстрый → быстрее = faster
• тёплый → теплее = warmer
• холодно → холоднее = colder

Here ещё means even or still more.

So:

• пасмурнее = cloudier
• ещё пасмурнее = even cloudier

It adds the idea that the weather was already gloomy, and then it became worse.

Compare:

• стало пасмурнее = it became cloudier
• стало ещё пасмурнее = it became even cloudier

Both are possible. This is mostly about word order and emphasis.

• начался ливень puts the focus first on the event starting
• ливень начался puts more focus on the downpour itself

Russian word order is more flexible than English. In this sentence, начался ливень sounds very natural in narration: first the weather got worse, and then a downpour started.

So the given order is not because of a special grammar rule; it is mainly a natural stylistic choice.

Because the verb is начаться, which means to begin / to start in the sense of to start on its own or to begin happening.

Compare:

• начать = to start something
• начаться = to begin, to start

Examples:

• Он начал урок. = He began the lesson.
• Урок начался. = The lesson began.

In your sentence:

• начался ливень = a downpour began / started

The rain is not starting something else; the rain itself is beginning, so начаться is the right verb.

This is a question of aspect.

• начаться is perfective
• начинаться is imperfective

Here the sentence describes a completed event: at some point, the downpour started. That is why Russian uses the perfective past:

• начался ливень = a downpour started

If you used начинался, it would sound more like was beginning or it would describe repeated/habitual starting in the right context.

So the perfective form is the normal choice for a single event in a narrative sequence.

Not quite. Ливень means a downpour, torrential rain, or heavy shower. It is stronger and more sudden than the general word дождь.

Compare:

• дождь = rain
• ливень = downpour / heavy rain

So начался ливень is stronger than начался дождь. It suggests a lot of rain started falling, often quite suddenly.

Because Russian usually does not use a dummy subject like English it for weather, time, and general conditions.

English says:

• It got dark
• It became colder
• It started raining

Russian often just uses the verb or an impersonal structure:

• стемнело
• похолодало
• стало пасмурнее

So стало ещё пасмурнее is fully complete in Russian even though there is no explicit subject.

Because the sentence contains two clauses joined by и:

1. После обеда стало ещё пасмурнее
2. начался ливень

Each clause has its own predicate:

• стало
• начался

When Russian joins two independent clauses with и, it normally uses a comma, just as English often would.

So the comma is there because this is essentially:

• After lunch it got even cloudier, and a downpour began.

It most directly means after lunch / after the midday meal. In many contexts, though, that naturally corresponds to in the afternoon.

So depending on the translation style, it could be rendered as:

• After lunch...
• Later in the afternoon...

But grammatically, the Russian phrase itself is literally after lunch / after the meal.

You can think of it as related to both, but in this sentence it functions like the comparative of the predicative/adverbial idea пасмурно = it is overcast / gloomy.

Compare:

• Сегодня пасмурно. = It is overcast today.
• Стало пасмурнее. = It became more overcast.

The dictionary adjective is пасмурный, but in weather expressions Russian very often uses the predicative form пасмурно and its comparative пасмурнее. So for this sentence, it is best to learn пасмурно → пасмурнее as a useful weather pattern.