На старой сумочке порвался ремешок, а на новой блузке появилась маленькая складка.

Because they come after на in the meaning of on or on the surface of, so Russian uses the prepositional case here:

• на старой сумочке
• на новой блузке

For feminine nouns ending in -а / -я, the prepositional singular often ends in -е:

• сумочка → на сумочке
• блузка → на блузке

The adjectives also change to match:

• старая → на старой
• новая → на новой

So the whole phrases mean on the old handbag and on the new blouse.

Because the adjectives must agree with the noun in gender, number, and case.

Here:

• сумочке = feminine singular, prepositional
• блузке = feminine singular, prepositional

So the adjectives also take the feminine singular prepositional ending -ой:

• старая сумочка → на старой сумочке
• новая блузка → на новой блузке

This is basic adjective-noun agreement in Russian.

Both are possible, but the version in your sentence is very natural because Russian often puts the location or setting first and then introduces what happened there.

So:

• На старой сумочке порвался ремешок
  = On the old handbag, the strap tore

This structure helps present the sentence as:

1. first, the place/object being discussed
2. then, the event that happened there

It sounds a bit like As for the old handbag, its strap tore.

If you say:

• Ремешок порвался на старой сумочке

that is still grammatical, but it focuses more directly on the strap.

Порваться is the verb to tear / to snap / to get torn. The ending -ся often makes a verb reflexive, but in many verbs it also creates a meaning like:

• to happen to itself
• to become
• to get into a state

So:

• рвать = to tear something
• порвать = to tear something completely / to tear off
• порваться = to tear, snap, come apart

In this sentence:

• ремешок порвался = the strap snapped / tore

This is a very natural way in Russian to describe something breaking or becoming damaged without naming an outside agent.

Russian often uses this pattern when something comes into existence or appears:

• появилась складка
• появилось пятно
• возникла проблема

This structure is very natural when introducing something new into the scene.

So:

• На новой блузке появилась маленькая складка
  literally means On the new blouse appeared a small crease

In normal English we usually say:

• A small crease appeared on the new blouse

Russian word order is flexible, and here the location is introduced first, then the event, then the thing that appeared.

Because Russian past tense verbs agree with the gender and number of the subject.

Here the subjects are:

• ремешок — masculine singular
• складка — feminine singular

So the past tense forms are:

• порвался for masculine
• появилась for feminine

Compare:

• ремешок порвался
• складка появилась
• пятно появилось — neuter
• складки появились — plural

This is one of the key differences from English, where past tense verbs usually do not change for gender.

Ремешок is a smaller or more specific kind of strap. It is related to ремень, but it has a diminutive / smaller-item feel.

Very roughly:

• ремень = belt, strap
• ремешок = little strap, narrow strap, fastening strap

For a handbag, ремешок is very natural, because it suggests the kind of strap a bag typically has.

So in this sentence, ремешок sounds more precise and idiomatic than ремень.

This is a question of aspect.

In your sentence, both verbs are perfective:

• порвался
• появилась

Perfective verbs present the event as a completed change or a single result:

• the strap snapped
• the crease appeared

If you used the imperfective forms:

• ремешок рвался
• складка появлялась

the meaning would be different. It could suggest:

• a repeated action
• a process
• background description
• something that kept happening or was in progress

So here the perfective is exactly right, because the sentence describes two completed incidents.

А often links two clauses while also showing a contrast or comparison.

Here the sentence compares two items:

• на старой сумочке ...
• а на новой блузке ...

So а works well because it feels like:

• while
• whereas
• and meanwhile
• and as for

It is not a strong opposition like but in every context, but it sets the two facts side by side.

If you used и, the sentence would sound more like a simple addition:

• The strap tore on the old handbag, and a small crease appeared on the new blouse

With а, there is a stronger sense of contrast between the two objects.

Yes. На is very natural when talking about something being on the surface of an object, especially clothing or accessories.

Examples:

• пятно на рубашке — a stain on a shirt
• дырка на носке — a hole in/on a sock
• пуговица на пальто — a button on a coat
• складка на блузке — a crease in/on a blouse

In English, we might sometimes say in or on depending on the noun, but Russian often uses на for visible features or defects on an item.

Because маленькая складка is the subject of the second clause.

The verb is:

• появилась

And what appeared?

• маленькая складка

Subjects are normally in the nominative case, so both words stay in their nominative forms:

• маленькая — feminine singular nominative
• складка — feminine singular nominative

The same is true in the first clause:

• ремешок is also nominative, because it is the subject of порвался.

Not very naturally. A very literal translation would be something like:

• On the old handbag snapped a strap, and on the new blouse appeared a small crease.

That is understandable, but it is not normal English word order.

A natural English version would usually be:

• The strap on the old handbag snapped, and a small crease appeared on the new blouse.

or

• The strap of the old handbag tore, and a small crease appeared on the new blouse.

So the Russian sentence is perfectly natural, but English usually prefers to put the subject earlier.

It depends on context. Порваться literally relates to tearing, but with a strap it can often be translated as:

• tore
• ripped
• snapped
• came apart

For ремешок, English often prefers snapped or broke, because a strap commonly breaks under tension.

So:

• ремешок порвался could be rendered as the strap tore or the strap snapped

The Russian focuses on the material separating or tearing; the best English choice depends on what sounds most natural in context.