Se vi gajnos tiom da mono kiom via fratino, vi ne ŝuldos al mi ion.

Because -os is the Esperanto ending for the future tense.

• gajnos = will earn
• ŝuldos = will owe

So the sentence is talking about a future possibility and its future result.

That is a normal difference between English and Esperanto.

In Esperanto, a future condition can use the future tense directly:

• Se vi gajnos... = If you earn / if you will earn...

English usually avoids will in the if-clause, but Esperanto does not have that restriction in the same way. So Se vi gajnos... vi ne ŝuldos... is perfectly natural.

tiom da is a quantity expression meaning that much / so much / as much depending on context.

Here:

• tiom = that much / so much
• da = a word used after quantity expressions
• mono = money

So tiom da mono means that much money or, in this sentence, as much money.

Because kiom is used for quantity, while kiel is usually used for manner, type, or degree/quality.

Here the sentence compares how much money two people earn, so Esperanto uses:

• tiom ... kiom = as much ... as

Compare:

• tiom da mono kiom via fratino = as much money as your sister
• tiel riĉa kiel via fratino = as rich as your sister

So kiom is the right choice because this is a comparison of amount.

Because after da, the noun normally does not take -n.

In tiom da mono:

• tiom da is the quantity expression
• mono stays in its basic form

Even though the whole phrase is the object of gajnos, the noun after da remains mono, not monon.

This is the same pattern as:

• multe da akvo = a lot of water
• iom da tempo = some time
• tiom da mono = that much money

Yes. The verb gajnos is understood and left out because it would just repeat what was already said.

So:

• tiom da mono kiom via fratino
  really means
• tiom da mono kiom via fratino gajnos

This kind of omission is very common in Esperanto, just as in English:

• as much money as your sister [earns]

Because it is not the direct object here.

In the understood full comparison:

• kiom via fratino gajnos

via fratino is the subject of the implied verb gajnos, so it stays without -n.

If it were a direct object, then it would take -n, but here it is the one doing the earning.

Because ŝuldi works like to owe someone something.

The usual pattern is:

• ŝuldi al iu ion = to owe someone something

So in this sentence:

• al mi = to me (the person owed)
• ion = something / anything (the thing owed)

Using min would wrongly make me the direct object.

Literally, ion is the accusative form of io, meaning something.

But in a sentence with ne, English often translates it as anything:

• vi ne ŝuldos al mi ion = you will not owe me anything

So the literal Esperanto structure is close to not owe me something, but the natural English meaning is not owe me anything.

Yes. That would also be correct.

These two mean basically the same thing:

• vi ne ŝuldos al mi ion
• vi ŝuldos al mi nenion

The important point is that Esperanto normally uses only one negative element in this kind of sentence. So if you use nenion, you normally do not also use ne.

Because the sentence begins with a conditional clause:

• Se vi gajnos tiom da mono kiom via fratino, ...

After that introductory if-clause, a comma is commonly used before the main clause:

• ..., vi ne ŝuldos al mi ion.

This works much like English punctuation in sentences beginning with if.