La cáscara del huevo fue al cubo de la basura.

Because del is the mandatory contraction of de + el.

• de el huevo → del huevo
• In standard Spanish, you normally must contract de + el to del.

So la cáscara del huevo literally breaks down as:

• la cáscara = the shell
• del huevo = of the egg

A similar contraction happens with a + el = al later in the sentence.

For the same reason: a + el contracts to al.

• a el cubo → al cubo

So:

• fue al cubo de la basura = it went to the rubbish bin / trash bin

This is one of the first contractions learners meet in Spanish:

• de + el = del
• a + el = al

Yes. Fue can be from either ser or ir.

In this sentence, it is from ir, because it is followed by a + place:

• fue al cubo de la basura

That pattern strongly signals movement toward a destination, so the meaning is went, not was.

Compare:

• Fue difícil = It was difficult. → from ser
• Fue al cubo de la basura = It went to the bin. → from ir

Context tells you which verb it is.

Because a is used for movement toward a destination, while en is used for location.

• fue al cubo de la basura = it went to the bin
• está en el cubo de la basura = it is in the bin

So:

• a = to
• en = in

This is the same basic contrast as in English.

Yes. Cáscara is the normal word for the outer shell or peel of many things, including an egg.

So:

• la cáscara del huevo = the eggshell

You will also see cáscara with fruit:

• la cáscara de naranja = orange peel
• la cáscara del plátano = banana peel

For an egg, cáscara is the usual everyday word.

It is understandable, but it sounds a little unusual or literary in everyday speech because the eggshell is treated as if it went somewhere by itself.

A native speaker would more naturally say something like:

• Tiré la cáscara del huevo al cubo de la basura. = I threw the eggshell in the bin.
• La cáscara del huevo acabó en el cubo de la basura. = The eggshell ended up in the bin.
• La cáscara del huevo fue a la basura. = The eggshell went in the rubbish / was thrown away.

So the original sentence is not wrong, but it is not the most everyday way to express the idea.

Because cubo de la basura is a normal noun phrase in Spanish meaning rubbish bin / trash bin.

Here de la basura means something like for rubbish or for the rubbish. Spanish often uses articles in places where English would not.

So although English says:

• the trash can
• the rubbish bin

Spanish often builds it as:

• the bin of the rubbish → el cubo de la basura

This is just a standard way the expression is formed.

Yes, but they are not always exactly the same.

In Spain:

• cubo de la basura = rubbish bin / trash bin, often a kitchen bin or household bin
• papelera = wastepaper basket / small bin, often in an office, bedroom, or public place
• contenedor = larger outside rubbish container

So if you mean the bin in the kitchen, cubo de la basura is a good choice. If you mean a small indoor bin, papelera may sound more natural.

Because fue is the preterite form, which presents the action as a completed event.

• fue = went

So the sentence describes a finished action: the eggshell ended up in the bin.

If you changed the tense, the meaning would shift:

• va al cubo de la basura = it goes to the bin / it is going to the bin
• iba al cubo de la basura = it was going to the bin / it used to go to the bin
• ha ido al cubo de la basura = it has gone to the bin

In the original, the event is seen as completed.

Because that is the topic or subject of the sentence: the thing the sentence is about.

Spanish word order is often flexible, but subject + verb + complement is very common:

• La cáscara del huevo = subject
• fue = verb
• al cubo de la basura = destination/complement

Putting La cáscara del huevo first makes it clear right away what went into the bin.

Yes. Very natural alternatives in Spain would be:

• Tiré la cáscara del huevo al cubo de la basura.
• Eché la cáscara del huevo al cubo de la basura.
• La cáscara del huevo acabó en el cubo de la basura.

They differ slightly:

• tiré = I threw away
• eché = I put/threw in
• acabó en = ended up in

If you want the most everyday phrasing, using a person as the subject is often the most natural choice.