Ŝi iras al ambasado pro tio, ke ŝia vizo eksvalidiĝis.

Because sia is a reflexive possessive and can only refer to the subject of its own clause.

In ke ŝia vizo eksvalidiĝis, the subject of that clause is vizo, not ŝi. So sia would be wrong there.

• ŝia vizo = her visa
• sia would have to refer back to the subject of the same clause, but the subject is vizo

So ŝia is the correct choice.

Yes. Pro tio, ke means because, more literally because of the fact that.

So:

• Ŝi iras al ambasado pro tio, ke ŝia vizo eksvalidiĝis.
• Ŝi iras al ambasado ĉar ŝia vizo eksvalidiĝis.

Both are understandable.
Ĉar is shorter and often more natural in everyday speech.

Pro tio, ke can sound a little more explicit or formal.

Because ke introduces a subordinate clause.

Esperanto often uses commas to separate subordinate clauses from the main clause, especially when a connector like ke, ĉar, se, or kiam begins the clause.

So in:

• pro tio, ke ŝia vizo eksvalidiĝis

the comma helps show that ke ŝia vizo eksvalidiĝis is a clause explaining the reason.

In Esperanto, the simple present tense often covers both ideas:

• goes
• is going

So Ŝi iras can mean either:

• She goes
• She is going

The context tells you which is meant. In this sentence, it most naturally means she is going.

Esperanto does not require a separate progressive form like English is going.

Because ambasado is the object of the preposition al.

After a preposition like al, you normally do not add the accusative -n:

• al ambasado = to an embassy / to the embassy

The -n is not needed because al already shows direction.

Compare:

• Ŝi iras al ambasado. = She is going to an embassy.
• Ŝi iras ambasaden. = She is going embassy-ward.
  This kind of directional -n is possible in Esperanto, but with al, you normally do not use it.

Esperanto has no indefinite article. There is no word corresponding to English a/an.

So:

• ambasado can mean an embassy
• vizo can mean a visa

Esperanto does have a definite article, la, when you mean the.

So depending on context, you might also see:

• Ŝi iras al la ambasado...
• ...ke la vizo eksvalidiĝis

In your sentence, the nouns are left without la, which is perfectly normal if they are not being marked as definite.

It is made from several parts:

• eks- = indicates cessation or former status; here it suggests no longer
• valid = valid
• -iĝ- = become, turn into
• -is = past tense

So eksvalidiĝis literally means something like:

• became no longer valid

In natural English, that is expired.

This is a very typical Esperanto word-building pattern.

Because the two actions happen at different times.

• iras = what she is doing now
• eksvalidiĝis = what happened earlier

The sentence means that her earlier visa expiration is the reason for her current trip to the embassy.

This is completely normal in Esperanto, just as in English:

• She is going to the embassy because her visa expired.

By itself, al ambasado is most literally to an embassy.

If the speaker means a specific embassy already known from context, many speakers would prefer:

• al la ambasado

However, Esperanto article use is sometimes a little broader or looser than English article use, and context matters a lot. So if the meaning has already been made clear, learners should not be surprised by article-less nouns.

Yes, ambasado is a normal and common word for embassy.

You may also encounter related forms such as:

• ambasadoro = ambassador
• ambasadejo = embassy building / place of an embassy

In many contexts, ambasado is enough and sounds natural.
If someone wants to emphasize the physical building, ambasadejo may appear, but ambasado is very common.

Ŝi is the third-person singular pronoun for she.

It is normally used for a female person. So:

• ŝi = she
• ŝia = her

In this sentence:

• Ŝi iras... = She is going...
• ŝia vizo = her visa

So the pronoun and the possessive adjective match in the expected way.