Al mia fratino mankas mono por nova biciklo.

With manki, the person who lacks something is often marked with al.

So Esperanto is structuring the idea more like:

• Money is lacking to my sister

rather than:

• My sister lacks money

That is why you get:

• al mia fratino = to my sister

This is a very common pattern with manki.

Mankas comes from manki, which means to be lacking, to be missing, or to be absent.

In this sentence, it means that something needed is not available. So:

• Al mia fratino mankas mono
  = My sister lacks money
  literally, Money is lacking to my sister

So the verb is not really focusing on what your sister has, but on what is missing.

Because mono is the grammatical subject of mankas, not the direct object.

In Esperanto, the -n ending usually marks a direct object. But here the structure is:

• mono mankas al mia fratino
  = money is lacking to my sister

So mono is the thing that is lacking, and that makes it the subject.

That is why it stays:

• mono, not monon

Yes. That is a very natural alternative.

• Mia fratino ne havas monon por nova biciklo
  = My sister doesn't have money for a new bicycle

The difference is mainly one of viewpoint:

• Al mia fratino mankas mono... emphasizes what is missing
• Mia fratino ne havas monon... emphasizes what she does not have

Both are correct and understandable. The version with mankas can sound slightly more like she is short of money.

Because por is a preposition, and after a preposition Esperanto normally uses the basic form, not -n.

So:

• por nova biciklo = for a new bicycle

The adjective agrees with the noun, so both stay without -n:

• nova biciklo

You would only use -n there in special cases, not in this normal prepositional phrase.

Esperanto has no indefinite article.

English distinguishes:

• a bicycle
• the bicycle

Esperanto only has a definite article:

• la

So:

• nova biciklo can mean a new bicycle
• la nova biciklo means the new bicycle

That is why por nova biciklo naturally means for a new bicycle.

Because the sentence is talking about money in a general sense, not some specific previously identified money.

• mono = money in general
• la mono = the money, some specific money

Here the meaning is simply that she lacks enough money, so mono without la is the natural choice.

Yes, Esperanto word order is fairly flexible.

For example, these are all possible:

• Al mia fratino mankas mono por nova biciklo.
• Mono mankas al mia fratino por nova biciklo.

The original order is perfectly natural. Starting with al mia fratino puts focus on your sister as the affected person.

Even when the word order changes, the roles stay clear because:

• al marks the person affected
• mono is still the subject

Here por means for, in the sense of for the purpose of getting/buying something.

So:

• mono por nova biciklo
  = money for a new bicycle

This does not necessarily mean the money is already set aside. It simply means the money needed in order to have or buy a new bicycle.

It is singular.

• fratino = sister
• fratinoj = sisters

So:

• mia fratino = my sister
• miaj fratinoj = my sisters

Here the sentence is about one sister.

Yes, very often that is exactly the nuance.

Mankas mono does not have to mean there is absolutely no money. It can also mean:

• there is not enough money
• some money is still needed
• money is short

So in context, the sentence may mean something like:

• My sister is short of money for a new bicycle
• My sister doesn’t have enough money for a new bicycle

That is one reason manki is useful: it often expresses shortage rather than total absence.