La pordon malferminta avino ridetas al ni.

It is made of several parts:

• ferm = close, shut
• mal- = the opposite of
• malfermi = to open
• -int- = active past participle
• -a = adjective ending

So malferminta means having opened or who opened.

In this sentence, it describes avino, so the idea is the grandmother who opened the door.

Because pordon is the direct object of malferminta.

Even though malferminta looks like an adjective, it still comes from a verb and can keep the verb’s object. The grandmother opened the door, so pordo gets the accusative ending:

• pordo = door
• pordon = the door as a direct object

A similar pattern would be:

• la libron leganta knabo = the boy reading the book

Here, libron is the object of leganta, just as pordon is the object of malferminta.

Because avino is the subject of the main verb ridetas.

The whole phrase la pordon malferminta avino is the subject of the sentence. In Esperanto, subjects normally do not take -n.

So:

• pordon = object of malferminta
• avino = subject of ridetas

If the grandmother were the object of another verb, then avino would take -n, and the participle would agree with it:

• Mi vidas la pordon malfermintan avinon.
  = I see the grandmother who opened the door.

Because it is acting as an adjective.

In Esperanto, participles can behave like adjectives, and adjectives end in -a. Here malferminta describes avino, so it has adjectival form.

It also agrees with the noun it modifies in number and case:

• malferminta avino = grandmother who opened
• malfermintaj avinoj = grandmothers who opened
• malfermintan avinon = grandmother who opened, as a direct object

So the -a is there because the word is modifying a noun.

Because the grandmother is the doer of the action, not the receiver.

• -int- = active past participle
  the noun did the action
• -it- = passive past participle
  the noun had the action done to it

So:

• malferminta avino = a grandmother who opened
• malfermita pordo = an opened door

If you said malfermita avino, that would mean an opened grandmother, which is obviously not what is meant.

Yes, essentially.

The participial phrase la pordon malferminta avino is a compact way to say:

• la avino kiu malfermis la pordon

So the sentence is basically the same as:

• La avino kiu malfermis la pordon ridetas al ni.

The participle version is shorter and more compressed. A learner will often find the relative clause easier at first, but both are correct Esperanto.

It shows that the opening happened before the smiling.

So the order of ideas is:

1. the grandmother opened the door
2. the grandmother smiles at us

It does not have to mean a long time before, only that the opening is already completed relative to ridetas.

Compare:

• malfermanta = opening, while opening
• malferminta = having opened, after opening
• malfermonta = about to open

Because rideti normally takes al for the person you smile at.

So:

• rideti al iu = to smile at someone

That is why the sentence has:

• al ni = at us / to us

Using nin would make we/us a direct object, which is not the normal pattern with rideti here.

Yes.

Esperanto allows some flexibility, so you can also say:

• La avino malferminta la pordon ridetas al ni.

That has the same basic meaning.

The original version puts the whole modifying phrase before avino:

• la pordon malferminta avino

That is perfectly correct, but it can feel a bit more compact or literary. Many learners find the version with the noun first easier to parse.

It comes from:

• ridi = to laugh
• -et- = a small or slight degree

So rideti literally suggests to laugh a little, but in normal usage it usually means to smile.

So:

• ridas = laughs
• ridetas = smiles

Yes, it is completely normal, but it can feel more condensed than a relative clause.

A native English speaker may find kiu malfermis la pordon easier at first, because it matches English structure more closely. But Esperanto uses participles very productively, and learners should get used to them.

So this sentence is not strange at all. It is just a more compact way to express the same idea.