La kurtenon malferminta avino ridetis al ni de la balkono.

Malferminta is the active past participle of malfermi (to open).

So:

• malfermi = to open
• malferminta = having opened / who had opened

In this sentence, it describes avino:

• La kurtenon malferminta avino = the grandmother who had opened the curtain

It works like a shortened relative clause in English.

Because kurtenon is the direct object of malferminta.

Even though malferminta is a participle, it still keeps the verb relationship of malfermi:

• someone opens something
• that something is the direct object

So:

• malfermi la kurtenon = to open the curtain
• la kurtenon malferminta avino = the grandmother having opened the curtain

The -n does not mean the curtain is the object of ridetis. It belongs to the participial idea having opened the curtain.

Because avino is the subject of the main verb ridetis.

The sentence is built around:

• avino ridetis = the grandmother smiled

Everything before avino just describes which grandmother:

• la kurtenon malferminta avino = the grandmother who had opened the curtain

So:

• avino = subject
• kurtenon = object of malferminta

Yes. It is essentially a shorter way to say:

• La avino, kiu malfermis la kurtenon, ridetis al ni de la balkono.

Or, depending on context:

• La avino, kiu estis malfermintа la kurtenon, ridetis al ni de la balkono.

In normal English, the smoothest translation is usually:

• The grandmother who had opened the curtain smiled at us from the balcony.

Esperanto often uses participles to make this kind of compact description.

Because malfermis is a finite verb, and this sentence already has the main finite verb ridetis.

You cannot normally say:

• La kurtenon malfermis avino ridetis...

That would give two main verbs without proper structure.

So Esperanto uses the participle malferminta to turn opened the curtain into a description of avino:

• the curtain-opening grandmother
• more naturally: the grandmother who had opened the curtain

The ending -inta shows:

• active meaning: the noun does the action
• past meaning: the action happened earlier than the main reference point

So malferminta avino means:

• the grandmother had opened the curtain
• the grandmother is the one who performed that action

Compare:

• malfermanta = opening
• malferminta = having opened / who had opened
• malfermonta = about to open / going to open

Here, malferminta shows that the curtain-opening happened before the smiling.

Al ni means to us or at us, depending on natural English style.

The verb rideti often takes al for the target of the smile:

• ridetis al ni = smiled at us

So:

• al marks the direction or recipient of the smile
• ni becomes al ni, not nin

This is similar to:

• paroli al iu = speak to someone
• skribi al iu = write to someone

Here de la balkono means from the balcony in the sense of from that location.

• de = from, off, away from, from the position of
• el = out of, out from inside

A balcony is not usually thought of as an enclosed interior space, so de la balkono is natural for from the balcony.

If something comes out of the inside of a room, box, or building, el is often better:

• el la domo = out of the house
• el la skatolo = out of the box

But:

• de la balkono = from the balcony

Most naturally, it describes ridetis:

• The grandmother smiled at us from the balcony.

So the idea is:

1. she had opened the curtain
2. she smiled at us
3. she did that from the balcony

In principle, readers use context and common sense to attach phrases like this. Here the most natural reading is that the smiling happened from the balcony.

Because all three nouns are presented as specific:

• la kurtenon = the curtain
• avino is made definite by the whole noun phrase: La kurtenon malferminta avino = the grandmother who had opened the curtain
• la balkono = the balcony

Esperanto uses la much like English the. If the speaker assumes the listener can identify the thing, la is used.

Yes, Esperanto word order is fairly flexible, but the given version is natural and clear.

This order:

• La kurtenon malferminta avino ridetis al ni de la balkono.

puts the full description of avino before the main verb.

You could also express the same idea with a relative clause:

• La avino, kiu malfermis la kurtenon, ridetis al ni de la balkono.

That may feel more familiar to an English speaker, but the original participial version is perfectly normal Esperanto.

Yes, basically.

• fermi = to close
• mal- = the opposite
• malfermi = to open

The prefix mal- is very common in Esperanto and forms opposites:

• bona = good → malbona = bad
• varma = warm → malvarma = cold
• fermi = close → malfermi = open

So malferminta literally contains the idea having un-closed, though in normal English we simply say having opened.