После падения у неё появилась царапина на ладони и синяк на колене.

Because после requires the genitive case.

• Dictionary form: падение = a fall / falling
• Genitive singular: падения

So:

• после падения = after the fall / after falling

This is a very common pattern in Russian: после + genitive.

It can mean either, depending on context.

Literally, падение is a noun meaning fall. So после падения is literally after the fall. But in natural English, this is often best translated as:

• after she fell
• after falling

Russian often uses a noun where English might prefer a clause or gerund.

Russian often expresses this idea differently from English.

Literally, у неё появилась царапина... is something like:

• at her, a scratch appeared...

In natural English, we say:

• she got a scratch
• she developed a bruise

So this is a very normal Russian way to talk about something appearing on or happening to someone.

Because у is a preposition, and here it takes the genitive case.

The pronoun она changes like this:

• nominative: она = she
• genitive after у: неё

So:

• у неё = at her / she has / on her

This construction is very common in Russian when talking about possession or something happening to a person.

After many prepositions, Russian third-person pronouns get an extra initial н-.

So:

• её → у неё
• ему → к нему
• ими → с ними

This is just a standard grammatical rule. So after у, её becomes неё.

Because in Russian, when the verb comes before a compound subject, it can agree with the nearest noun.

Here the first noun after the verb is:

• царапина — feminine singular

So the verb appears as:

• появилась — feminine singular past tense

This is normal Russian. You may also see plural agreement in similar sentences, but singular agreement with the nearest noun is very common when the verb comes first.

Because they are the subject(s) of the verb появилась.

Russian is not literally saying she got a scratch and a bruise with scratch and bruise as direct objects. Instead, it is saying that a scratch and a bruise appeared.

So:

• царапина = nominative
• синяк = nominative

They are the things that appeared.

Because на here means on, and when it shows location, it normally takes the prepositional case.

• ладонь → на ладони
• колено → на колене

So:

• царапина на ладони = a scratch on the palm
• синяк на колене = a bruise on the knee

Because Russian often leaves out possessive words like her, his, or my when the owner is obvious from context, especially with body parts.

So:

• на ладони can mean on the palm / on her palm
• на колене can mean on the knee / on her knee

In this sentence, since у неё already tells us we are talking about her, Russian does not need to repeat that idea.

Yes, you could, but it is a little different in nuance.

• она получила царапину и синяк = she got a scratch and a bruise
• у неё появилась царапина и синяк = a scratch and a bruise appeared on her / she developed a scratch and a bruise

The version with появилась sounds like a neutral description of the result.
The version with получила sounds more like received/got those injuries.

Both can work, but the original sentence is very natural.

Because и is simply joining two parallel nouns:

• царапина на ладони
• синяк на колене

This is a normal coordination with and, so no comma is needed.

The sentence is arranged roughly like this:

• После падения — time/background
• у неё — person affected
• появилась — verb
• царапина на ладони и синяк на колене — what appeared

Russian word order is flexible, so parts can move for emphasis. But this version sounds natural and neutral.

It presents the situation smoothly:

1. first the event (after the fall),
2. then the person (with her),
3. then the result (a scratch... and a bruise...).