Лучше бы я взял зонтик, тогда моя куртка не стала бы мокрой.

Yes, лучше бы + past tense is a very common way to express regret about the past, often close to English I should have... or It would have been better if...

So:

Лучше бы я взял зонтик
= I should have taken an umbrella
literally more like
= It would have been better if I had taken an umbrella

A few important points:

• лучше by itself means better
• бы adds the hypothetical/regret meaning
• the verb usually appears in the past tense form

So even though взял looks like a simple past form, with лучше бы the whole phrase expresses an unreal past situation.

Бы is the particle that helps form the conditional or hypothetical meaning in Russian.

In this sentence, it appears in both parts because both parts are unreal/hypothetical:

• Лучше бы я взял зонтик
• тогда моя куртка не стала бы мокрой

The first part expresses regret: it would have been better if...
The second part expresses the imagined result: then my jacket would not have become wet

Russian often uses бы in each clause that is hypothetical. That is very normal.

Compare:

• Если бы я взял зонтик, куртка не стала бы мокрой.
• Лучше бы я взял зонтик, тогда моя куртка не стала бы мокрой.

So the repeated бы is not a mistake; it is standard grammar.

In Russian, hypothetical past situations are usually built with:

• a past tense verb form
• plus бы

So взял бы or лучше бы взял does not mean plain took; it means would have taken or should have taken, depending on context.

This is one of the most important things for English speakers to get used to:

• Russian does not have a special conditional verb form like English would take
• instead, it uses the past tense form + бы

So:

• я взял = I took
• я бы взял = I would take / would have taken
• лучше бы я взял = I should have taken / it would have been better if I had taken

Because взять is perfective, and here the speaker is talking about a single completed action: taking an umbrella.

• брать = imperfective
• взять = perfective

In this sentence, the idea is:

I should have taken an umbrella
meaning one concrete action, with a result.

That is why взял is natural.

If you used брал, it would sound odd here, because imperfective would suggest something ongoing, repeated, or less focused on completion.

Russian often uses perfective verbs in unreal past situations when the speaker is imagining a completed action and its result.

Зонтик is in the accusative case because it is the direct object of взял:

• взял что? → зонтик

Since зонтик is an inanimate masculine noun, its accusative singular looks the same as the nominative singular:

• nominative: зонтик
• accusative: зонтик

So the form does not change, but the case is still accusative.

If it were an animate masculine noun, the accusative would often look different.

This is a very common question.

Стать means to become, while быть means to be.

So:

• не стала бы мокрой = would not have become wet
• не была бы мокрой = would not be / would not have been wet

In this context, стала бы is very natural because the jacket was originally dry and then changed state. The sentence focuses on that change:

dry → wet

That is why стать мокрой fits well.

If you said не была бы мокрой, it would focus more on the state rather than the change. It is not impossible, but не стала бы мокрой sounds more idiomatic here.

Because after стать, Russian normally uses the instrumental case for the noun or adjective that describes the new state.

So:

• стать мокрой
• стать врачом
• стать счастливым

Here, куртка is feminine singular, so the adjective in the instrumental feminine singular is:

• мокрая = nominative
• мокрой = instrumental

That is why we get:

моя куртка не стала бы мокрой

This is a very important pattern:

• быть + adjective often uses nominative
  
  • куртка была мокрая
• стать + adjective usually uses instrumental
  
  • куртка стала мокрой

Yes, Russian could sometimes leave it out if the context were very clear, but here моя куртка is perfectly natural and helps make the result explicit.

The speaker is saying:

• I should have taken an umbrella
• then my jacket would not have gotten wet

Including моя adds clarity and makes the sentence feel complete. It can also sound slightly more personal or concrete.

In conversation, if the context were obvious, a speaker might say something shorter, but this full version is very normal.

Yes. A very natural alternative would be:

Если бы я взял зонтик, моя куртка не стала бы мокрой.

This means:

If I had taken an umbrella, my jacket would not have gotten wet.

The difference is mainly in emphasis:

• Лучше бы я взял зонтик... emphasizes regret
• Если бы я взял зонтик... presents a more neutral conditional idea

The original sentence starts with the regret first, which is why it feels closer to I should have taken an umbrella.

Тогда here means something like then or in that case.

It connects the imagined action with its imagined result:

• Лучше бы я взял зонтик
• тогда моя куртка не стала бы мокрой

So it signals:

If that had happened, then this would have followed.

It is not absolutely required, but it makes the logical connection clearer and more natural.

Without it, the sentence would still work:

Лучше бы я взял зонтик, моя куртка не стала бы мокрой.

But тогда helps structure the thought more clearly.

Yes, the position of бы is somewhat flexible, but not completely free.

In your sentence:

• Лучше бы я взял зонтик
• моя куртка не стала бы мокрой

this placement sounds natural.

A few useful points:

1. Бы often comes after the first stressed or important word in the clause.
2. It can move around a little depending on style and emphasis.
3. It still has to stay connected to the clause it belongs to.

For example, these can all be possible in different contexts:

• Лучше бы я взял зонтик
• Я бы лучше взял зонтик
  but this can sound more like I’d rather take an umbrella, depending on context
• Куртка бы не стала мокрой
• Куртка не стала бы мокрой

So word order matters for nuance. In your sentence, the chosen order is natural for expressing regret about a past event.