У сумочки сломался ремешок, поэтому я не беру её на работу.

Russian very often uses у + genitive to show possession in sentences like this, especially when talking about a part of something.

So У сумочки сломался ремешок is literally something like:

• At the handbag, the strap broke

But naturally it means:

• The handbag’s strap broke
• The purse has a broken strap

This pattern is extremely common in Russian:

• У машины сломалось колесо = The car’s wheel broke
• У мальчика болит голова = The boy has a headache

So even though it sounds unusual from an English point of view, it is very normal Russian.

Because у requires the genitive case.

The dictionary form is сумочка.
Its singular genitive form is сумочки.

So:

• сумочка = handbag / purse
• у сумочки = of the handbag / at the handbag

This is purely a case change caused by the preposition у.

Yes. Сумочка is a diminutive form of сумка.

Very roughly:

• сумка = bag
• сумочка = small bag / handbag / purse

But in real Russian, diminutives do not always mean physically small. They can also sound:

• more everyday
• more natural in conversation
• slightly softer or more personal

So in many contexts, сумочка is simply the normal word for a woman’s handbag or purse.

Because the verb agrees with ремешок, not with сумочка.

In this sentence, the grammatical subject is ремешок:

• ремешок = strap

And ремешок is masculine singular, so in the past tense the verb must also be masculine singular:

• сломался = broke

Compare:

• ремешок сломался = the strap broke
• ручка сломалась = the handle broke
• колесо сломалось = the wheel broke

So the gender of the verb follows the thing that actually broke.

Here -ся helps make the verb intransitive: it means that the thing itself broke / became broken, rather than someone breaking it.

Compare:

• Я сломал ремешок = I broke the strap
• Ремешок сломался = The strap broke

So in this sentence:

• сломался = broke / became broken

It does not necessarily mean the strap broke itself on purpose; it is just the normal Russian way to say that something ended up broken.

Because ремешок is the subject of the verb сломался.

The strap is the thing that performed the grammatical action of breaking.

So the structure is:

• у сумочки = possessor, in genitive
• ремешок = subject, in nominative
• сломался = past-tense verb agreeing with ремешок

That is why Russian does not put ремешок in the genitive here.

Because the Russian imperfective present can often cover meanings like:

• I don’t take
• I’m not taking
• I’m not taking it these days / in this situation

So я не беру её на работу can mean that, because the strap is broken, the speaker is currently not taking the bag to work.

Why imperfective?

• брать is the imperfective verb
• беру is its 1st person singular present form

It sounds natural for a current or repeated practical decision.

A perfective future form like не возьму would sound more like:

• I won’t take it
  for one specific future occasion.

Because её is the accusative form of она for a feminine singular direct object.

It refers back to сумочку:

• я не беру её = I’m not taking it

Since сумочка is feminine, the pronoun is feminine too.

Also, in normal writing you will often see ее without the dots, but it is pronounced её.

Because Russian uses the fixed expression на работу for to work when talking about going to one’s workplace.

Examples:

• идти на работу = to go to work
• ехать на работу = to go to work by transport
• брать что-то на работу = to take something to work

Here на takes the accusative because there is movement toward a destination:

• на работу = to work

Compare:

• на работе = at work

So:

• Я не беру её на работу = I don’t take it to work
• Она лежит у меня на работе = It is at my workplace

Yes, Russian word order is flexible.

This order is natural because it introduces the bag first as the topic, and then gives the new information: the strap broke.

Possible variants include:

• У сумочки сломался ремешок
• Ремешок у сумочки сломался

Both are possible, but they sound slightly different in emphasis.

The given version is a very natural way to say:

• As for the purse, its strap broke

So the word order is not random; it helps organize what is already known and what is new information.

Yes, поэтому means therefore, that’s why, or so.

In this sentence it links cause and result:

• У сумочки сломался ремешок = The strap broke
• поэтому я не беру её на работу = so I don’t take it to work

The comma is used because the sentence has two parts:

1. the cause
2. the result

So the punctuation is normal Russian punctuation for this kind of connection.

Not necessarily.

Both words contain suffixes that historically look diminutive:

• сумочка
• ремешок

But in everyday Russian, such forms are often just the normal, most natural words for the object.

So in translation, English usually does not need to copy the diminutive idea:

• сумочка → purse / handbag
• ремешок → strap

Only in some contexts would you want to emphasize little.