A pesar de que la piedra estaba resbaladiza, mi sobrina cruzó el arroyo con cuidado.

A pesar de que means despite the fact that or even though.

It introduces a contrast:

• A pesar de que la piedra estaba resbaladiza...
• Even though the stone was slippery...

It is a fixed expression:

• a pesar de
  • noun/infinitive
    
    • A pesar de la lluvia = despite the rain
    • A pesar de ser joven = despite being young
• a pesar de que
  • verb
    
    • A pesar de que llovía... = even though it was raining...

It is quite common and sounds natural in both spoken and written Spanish.

Because estaba is the imperfect, and here it describes the background condition of the stone.

• La piedra estaba resbaladiza = the stone was slippery
  This sets the scene and describes an ongoing state.
• Mi sobrina cruzó el arroyo = my niece crossed the stream
  This is the main completed action.

Spanish often uses:

• imperfect for descriptions, states, background information
• preterite for completed actions

So the sentence naturally combines them:

• background: estaba
• main event: cruzó

Cruzó is the preterite, which is used for a completed action.

Here, the sentence presents the crossing as one finished event:

• mi sobrina cruzó el arroyo = my niece crossed the stream

If you said cruzaba, it would sound more like:

• she was crossing the stream
• or she used to cross the stream

That would change the meaning. In this sentence, the speaker is talking about one specific completed crossing, so cruzó is the right choice.

Because resbaladiza is an adjective that agrees with la piedra, which is feminine singular.

Agreement in Spanish:

• masculine singular: resbaladizo
• feminine singular: resbaladiza
• masculine plural: resbaladizos
• feminine plural: resbaladizas

Since piedra is feminine singular, the adjective must match:

• la piedra resbaladiza

In this sentence, estaba is in the indicative because the speaker treats the stone being slippery as a real, known fact.

• A pesar de que la piedra estaba resbaladiza... = even though the stone was slippery

With expressions like a pesar de que, Spanish can use:

• indicative when the information is presented as factual
• subjunctive when it is uncertain, hypothetical, or not being asserted as fact

Compare:

• A pesar de que estaba resbaladiza... = even though it was slippery
• A pesar de que estuviera resbaladiza... = even if it was slippery / despite it possibly being slippery

In your sentence, the indicative is the normal choice.

Arroyo usually means stream, brook, or sometimes creek.

It is smaller than a río (river).

So:

• cruzó el arroyo = crossed the stream/brook

Depending on context, English could translate it slightly differently, but stream is a very safe choice here.

Con cuidado means carefully or more literally with care.

Spanish often uses a prepositional phrase where English uses an adverb:

• con cuidado = carefully
• con calma = calmly
• con dificultad = with difficulty

You could also say cuidadosamente, and it would be grammatically correct:

• Mi sobrina cruzó el arroyo cuidadosamente

But con cuidado is often more common and natural in everyday Spanish.

Because Spanish normally does not use an article before a possessive adjective like mi, tu, su, nuestro, etc.

So you say:

• mi sobrina = my niece
• mi hermano = my brother
• su casa = his/her house

Not:

• la mi sobrina ❌

The possessive already identifies the noun, so no article is needed.

Yes. Piedra can mean stone or rock, depending on context.

In this sentence, the stone is a natural translation, but rock could also work in some contexts.

Spanish does not always separate stone and rock the same way English does. So:

• la piedra estaba resbaladiza = the stone was slippery
• possibly the rock was slippery, depending on the situation

They have accents for different reasons.

Cruzó

• The accent marks the correct stress: cru-ZÓ
• It also helps identify the form as third person singular preterite
  • cruzo = I cross
  • cruzó = he/she crossed

So the accent is very important for meaning.

Resbaladiza

• The written accent is not on resbaladiza, because it follows normal stress rules
• So actually resbaladiza has no written accent mark

In your sentence, the only written accent mark is in cruzó.

Yes, in many situations you could say:

• Aunque la piedra estaba resbaladiza, mi sobrina cruzó el arroyo con cuidado.

This also means although/even though the stone was slippery, my niece crossed the stream carefully.

The difference is small here:

• aunque is very common and flexible
• a pesar de que can sound a bit more emphatic, like despite the fact that

Both are correct and natural.

The given word order is natural, but Spanish is fairly flexible.

Original:

• A pesar de que la piedra estaba resbaladiza, mi sobrina cruzó el arroyo con cuidado.

You could also say:

• Mi sobrina cruzó el arroyo con cuidado, a pesar de que la piedra estaba resbaladiza.

Both are correct. The original version puts the contrast first, which makes the slippery stone feel like the important setup before the action happens.