La jabonera del baño se cayó al suelo, pero no se rompió.

Because jabonera is a different noun from jabón.

• el jabón = the soap
• la jabonera = the soap dish / soap holder

A noun’s gender in Spanish belongs to the noun itself, not automatically to the word it comes from. Also, nouns ending in -era are very often feminine.

Because de + el contracts to del in Spanish.

So:

• de el baño → del baño

This is a normal mandatory contraction.
La jabonera del baño literally means the soap dish of the bathroom, but in natural English it is more like the bathroom soap dish or the soap dish in the bathroom.

Literally, it is of the bathroom, but that literal wording is not how English normally says it.

In this sentence, del baño identifies which soap dish we mean: the one associated with the bathroom. In natural English, that usually becomes:

• the bathroom soap dish
• the soap dish in the bathroom

So it is mainly a way of specifying location or association.

Both caer and caerse can refer to falling, but caerse is very common when something or someone falls down in an actual incident, often unexpectedly or accidentally.

So:

• cayó = fell
• se cayó = fell down / ended up falling

In everyday Spanish, se cayó al suelo sounds very natural for something that accidentally fell onto the floor.

The se here usually is not translated as itself. It is just part of the way this verb is often used.

Al suelo means to the floor or onto the floor.

It uses al because a + el contracts to al:

• a el suelo → al suelo

Spanish often uses a to show movement toward a place:

• ir al baño = to go to the bathroom
• caerse al suelo = to fall to the floor

So al suelo is the natural expression here.

Because al suelo focuses on movement toward the floor.

• se cayó al suelo = it fell to the floor
• está en el suelo = it is on the floor

So:

• a / al = direction or result of movement
• en = location

Here the sentence describes the action of falling, so al suelo is the best choice.

Because romper and romperse work differently.

• romper = to break something
  
  • Juan rompió el vaso = Juan broke the glass
• romperse = to break / to get broken
  
  • El vaso se rompió = The glass broke

In your sentence, the soap dish is the thing that might have become broken, so Spanish uses romperse:

• no se rompió = it didn’t break

If you said no rompió, it would sound incomplete, because romper normally needs an object: didn’t break what?

They are similar in form, but not exactly identical in function.

• In se cayó, caerse is a very common pronominal way to say to fall down
• In se rompió, romperse means to break / become broken

In both cases, se is not usually translated literally. It is part of the verb pattern that Spanish uses for these kinds of events.

A useful way to think about it is:

• se cayó = it fell
• se rompió = it broke

Because the sentence describes completed events in the past:

• the soap dish fell
• it did not break

The preterite is the normal tense for single finished actions like these.

Compare:

• se cayó = it fell
• se rompió = it broke
• se caía = it was falling / used to fall
• se rompía = it was breaking / used to break

The imperfect would not fit as well here because this sentence is about one specific completed incident.

Because Spanish often leaves out subject pronouns when they are clear from context.

English needs it:

• it didn’t break

Spanish usually does not:

• no se rompió

Here, it is clear that the subject is still la jabonera, so Spanish does not need to repeat ella.

In Spain, suelo is a very common word for floor or ground, and caerse al suelo is a very natural expression.

Piso can also mean floor in some contexts, but in Spain it very often means apartment/flat. Because of that, al suelo is the safest and most idiomatic choice here.

So for this sentence, al suelo sounds exactly right.