Se a menina não tivesse caído da bicicleta, o cotovelo não teria ardido tanto.

It is a past unreal conditional sentence.

It talks about:

• something that did happen in the past: the girl fell off the bike
• and an imagined different result: her elbow would not have hurt so much

In English, this is like:

• If the girl hadn’t fallen off the bicycle, her elbow wouldn’t have hurt so much.

Portuguese usually builds this pattern with:

• se + pluperfect subjunctive
• conditional perfect in the result clause

So here:

• se ... tivesse caído
• não teria ardido

Because Portuguese uses the pluperfect subjunctive in this kind of hypothetical past if clause.

Tivesse caído breaks down like this:

• tivesse = imperfect subjunctive form of ter
• caído = past participle of cair

Together, tivesse caído means had fallen in a hypothetical or unreal sense.

After se, this is the standard pattern for a past situation that did not happen the way the speaker is imagining it:

• Se eu tivesse estudado... = If I had studied...
• Se ela tivesse chegado antes... = If she had arrived earlier...

Because não caiu is just the simple past. It refers to a real past event, not an unreal or hypothetical one.

Compare:

• Se a menina não caiu da bicicleta...
  This sounds odd for this meaning, because it does not match the idea of a counterfactual past.

• Se a menina não tivesse caído da bicicleta...
  This correctly means if the girl had not fallen off the bicycle.

So if you want to express an imagined alternative to what really happened, Portuguese normally uses tivesse caído, not caiu.

Teria ardido is the conditional perfect.

It is formed with:

• teria = conditional of ter
• ardido = past participle of arder

It corresponds to English would have hurt / would have burned / would have stung.

So:

• não teria ardido tanto = would not have hurt so much

This tense is used for the result in a past unreal conditional:

• Se eu tivesse ido, teria visto você.
• If I had gone, I would have seen you.

Because when a past participle is used with ter, it normally does not agree with the subject in gender or number.

So:

• o cotovelo teria ardido
• as mãos teriam ardido
• ela teria ardido por dentro
  all keep the participle in the same form: ardido

This is different from some adjective-like uses of participles, but with ter + participle, agreement usually does not happen.

Portuguese often uses the definite article with body parts when the owner is obvious from context.

So:

• o cotovelo here naturally means her elbow

This is very common in Portuguese:

• Ela levantou a mão. = She raised her hand.
• Ele machucou o joelho. = He hurt his knee.

You can say o cotovelo dela if you want to emphasize whose elbow it was, but it is not necessary here.

Because cair da bicicleta means to fall off the bicycle.

• da = contraction of de + a
• literally, it suggests movement from the bicycle

This is different from de bicicleta, which usually means by bicycle or on a bicycle in a general way:

• Ir de bicicleta = to go by bicycle
• Andar de bicicleta = to ride a bicycle

So:

• cair da bicicleta = fall off the bicycle
• andar de bicicleta = ride a bicycle

Yes, arder literally means to burn, but in Portuguese it is also commonly used for the sensation of:

• stinging
• smarting
• burning pain
• irritation

So for a scraped elbow, arder is very natural.

Examples:

• Meu joelho está ardendo. = My knee is stinging/burning.
• O corte ardeu muito. = The cut hurt/stung a lot.

In this sentence, ardido suggests that the elbow stung/burned/hurt, probably because of the scrape from the fall.

Tanto means so much, that much, or so badly, depending on context.

Here it modifies ardido:

• não teria ardido tanto = wouldn’t have hurt so much

It intensifies the result.

Compare:

• não teria ardido = it wouldn’t have hurt
• não teria ardido tanto = it wouldn’t have hurt so much

Yes. Portuguese allows both orders.

You can say:

• Se a menina não tivesse caído da bicicleta, o cotovelo não teria ardido tanto.
• O cotovelo não teria ardido tanto se a menina não tivesse caído da bicicleta.

Both mean the same thing.

The difference is mainly one of focus:

• starting with Se... foregrounds the condition
• starting with O cotovelo... foregrounds the result

The accent shows that the vowels are pronounced in separate syllables:

• ca-í-do

Without the accent, a learner might expect a different pronunciation pattern. The accent marks the stressed í and helps show the hiatus.

This happens in many similar forms:

• saído
• caído
• traído

So caído is the past participle of cair, and the accent is part of the correct spelling.

Yes. The sentence is already natural, but in everyday Brazilian Portuguese you might also hear slightly different wording, such as:

• Se a menina não tivesse caído da bicicleta, o cotovelo dela não ia arder tanto.
• Se ela não tivesse caído da bicicleta, o cotovelo não teria ardido tanto.

A few notes:

• não teria ardido is a bit more formal or written
• não ia arder is very common in speech
• dela may be added for clarity, though it is not required

So the original sentence is correct and natural, especially as a good model of standard grammar.