Il mocio resta in balcone finché la candeggina non si asciuga del tutto.

Resta comes from restare, which means to remain or to stay. It gives the idea that the mop continues to be on the balcony until something changes.

Stare can sometimes work too, but restare often sounds more natural when you mean it stays there until...

Yes. Sul balcone is very common and may sound more neutral or textbook-like to many learners.
In balcone is also used by many speakers and is understood as out on the balcony.

So in this sentence, the meaning does not really change.

Not really. Italian prepositions do not always match English ones word for word.

Here, in balcone means something like out on the balcony, not inside the balcony. It is best to learn it as a location expression rather than translating in too literally.

Here finché means until.

But in other contexts, finché can also mean as long as or while. Context tells you which meaning is intended.

For example:

• Finché piove, resto a casa = As long as it rains, I stay home
• Aspetta finché arriva Marco = Wait until Marco arrives

This is one of the most common learner questions.

In Italian, finché non often simply means until. The non here is not a real negative. It is often called an expletive or pleonastic non.

So:

finché la candeggina non si asciuga del tutto
means
until the bleach dries completely

It does not mean until the bleach does not dry.

Because Italian often uses a pronominal or reflexive-looking form where English does not.

Compare:

• asciugare qualcosa = to dry something
• asciugarsi = to dry / to become dry / to dry out

So la candeggina si asciuga means the bleach dries or dries out.

English does not need itself here, but Italian commonly uses si with verbs like this.

Not exactly in meaning.

With si lava, the subject is clearly doing something to itself: he washes himself.

With si asciuga in this sentence, the meaning is closer to becomes dry or dries out. So it looks reflexive, but in practice it often works like an intransitive or pronominal verb rather than a strongly literal reflexive.

Italian often uses the present tense for:

• general truths
• routine actions
• instructions
• descriptions of what normally happens

This sentence sounds like a general procedure or rule: the mop stays on the balcony until the bleach is fully dry.

So the present tense is perfectly natural.

Del tutto means completely, entirely, or totally.

So:

si asciuga del tutto = dries completely

It strengthens the idea that the bleach must be fully dry, not just partly dry.

Because candeggina is a feminine noun in Italian, so it takes la.

Grammatical gender in Italian does not always match anything logical in English, so this is something you simply learn with the noun:

• la candeggina = bleach

Even though the sentence is about a substance, Italian still treats candeggina as a normal feminine singular noun.