Spužva je ostala pokraj sudopera, pa sam brzo obrisala stol.

Because Croatian normally uses the perfect tense for past events:

• je = auxiliary verb from biti (is/has)
• ostala = past participle of ostati (to remain / stay)

So je ostala means remained / stayed in the past. In standard Croatian, 3rd person past tense usually includes the auxiliary.

Because it agrees with spužva, which is a feminine singular noun.

In Croatian, past participles agree with the subject in gender and number:

• masculine: ostao
• feminine: ostala
• neuter: ostalo

Since spužva is feminine, the sentence uses ostala.

ostati means to stay or to remain, so it suggests that the sponge was left there or remained there.

• Spužva je bila pokraj sudopera = The sponge was next to the sink
• Spužva je ostala pokraj sudopera = The sponge stayed/remained next to the sink

So ostala adds the idea of remaining in that place, not just being there.

pokraj means next to, beside, or by.

It is very similar to:

• kraj
• pored

All of these can mean that something is located near something else. In everyday speech, pored is especially common.

Because pokraj requires the genitive case.

The noun changes like this:

• nominative: sudoper
• genitive: sudopera

So:

• pokraj sudopera = next to the sink

This is a common thing in Croatian: certain prepositions require a specific case.

Here pa means something like:

• so
• and so
• and then

It links the first clause to the next one and suggests a result or next step.

So the sentence flow is roughly:

• The sponge remained next to the sink, so I quickly wiped the table.

Because Croatian often omits subject pronouns when they are already clear from the verb form.

In sam obrisala, the word sam shows 1st person singular, so ja is not necessary.

You could say pa sam brzo obrisala stol without ja, and it sounds completely natural.
Adding ja would usually give extra emphasis.

In this sentence, sam is not the word meaning alone.

Here sam is the 1st person singular auxiliary of biti, used to form the past tense:

• sam obrisala = I wiped

It is true that sam can also mean alone, but that is a different use. In this sentence, it is purely grammatical.

Because the speaker is female.

In Croatian past tense, the participle agrees with the subject's gender:

• male speaker: obrisao
• female speaker: obrisala

So sam obrisala tells you that the person speaking is a woman.

Because sam is a clitic: a short, unstressed word that usually appears very early in the clause.

After a connector like pa, it is very natural to say:

• pa sam brzo obrisala stol

Croatian word order is flexible, but clitics such as sam and je tend to stay near the beginning of their clause.

Because stol is the direct object of obrisala, so it is in the accusative case.

For many masculine inanimate nouns, the accusative singular looks the same as the nominative singular:

• nominative: stol
• accusative: stol

So the case has changed, but the form stays the same.

Yes. obrisati is a perfective verb, so it presents the action as completed.

• obrisala = she wiped it, finished wiping it
• brisala = she was wiping / used to wipe / was in the process of wiping

So obrisala stol means the table got wiped, as a finished action.

brzo is an adverb meaning quickly.

Croatian word order is fairly flexible, so adverbs can often move depending on style or emphasis.
Here, sam brzo obrisala stol is a very natural order.

You might also hear:

• brzo sam obrisala stol

Both are possible, with slightly different emphasis and rhythm.

Because Croatian does not have articles.

So:

• spužva can mean a sponge or the sponge
• sudoper can mean a sink or the sink
• stol can mean a table or the table

The exact meaning is understood from context.