Acaricia al gato despacio para que no tenga miedo.

Yes. Acaricia here is the informal singular command (tú) of acariciar = to stroke / to pet / to caress.

So the sentence is addressing one person directly:

• acaricia = stroke / pet
• implied subject = you

If you wanted a more formal command (usted), you would say:

• Acaricie al gato despacio...

And if you were speaking to several people in Spain (vosotros), you would say:

• Acariciad al gato despacio...

Because Spanish often uses the personal a before a specific direct object when it refers to a person or sometimes an animal, especially when the animal is seen as individual or familiar.

Here, el gato is a specific cat, so Spanish uses:

• acaricia al gato

Also, a + el = al, so:

• a el gato → al gato

This does not mean to the cat here. It is just the required a before the direct object.

Not always, but it is very common when the animal is specific, known, or treated as an individual.

Compare:

• Busco un gato. = I’m looking for a cat.
  
  • more general, non-specific
• Busco al gato. = I’m looking for the cat.
  
  • specific cat

With pets and familiar animals, Spanish often uses the personal a naturally.

Despacio means slowly or gently, depending on context. Here it suggests petting the cat gently/slowly so it does not get scared.

It is an adverb, and Spanish is fairly flexible about where it goes. These are all possible, though they may sound slightly different in emphasis:

• Acaricia al gato despacio.
• Acaricia despacio al gato.
• Despacio, acaricia al gato. (more marked, more like instruction)

The version in your sentence is very natural.

Because the subject changes.

• you pet the cat
• the cat should not be afraid

When Spanish uses para que and the subject of the second part is different, it normally takes a conjugated verb in the subjunctive:

• Acaricia al gato despacio para que no tenga miedo.

If the subject stays the same, Spanish often uses para + infinitive instead:

• Habla despacio para no cometer errores.
  = Speak slowly so as not to make mistakes.

In your sentence, the cat is the one who might feel fear, so para que + subjunctive is needed.

Because para que introduces a clause of purpose: you do one thing so that something else happens or does not happen.

After para que, Spanish normally uses the subjunctive:

• para que tenga
• para que no tenga

Here, tenga is the present subjunctive of tener.

This is very common:

• Te lo explico para que lo entiendas.
• Cierra la puerta para que no entre frío.
• Acaricia al gato despacio para que no tenga miedo.

Because tener miedo is the most natural everyday expression for to be afraid or to feel fear.

Literally, it is to have fear, but in normal English we translate it as:

• to be afraid
• to get scared
• to feel fear

So:

• no tenga miedo = so that it isn’t afraid / doesn’t get scared

Spanish does have the verb temer (to fear), but it is usually less everyday in this kind of sentence:

• para que no tema is grammatical, but less natural here than para que no tenga miedo

Yes, if the context already makes it clear that lo refers to the cat.

• Acarícialo despacio para que no tenga miedo.

This means the same thing: Pet him/it gently so that he/it doesn’t get scared.

Notice the accent mark in acarícialo. When object pronouns are attached to an affirmative command, Spanish often adds an accent to keep the normal stress.

So both are fine:

• Acaricia al gato despacio... = more explicit
• Acarícialo despacio... = more compact, once the cat is already known in context

Because Spanish usually omits subject pronouns when they are clear from the verb form or the situation.

• Acaricia already tells you this is a tú command
• so tú is unnecessary

You could say:

• Tú, acaricia al gato despacio...

but that sounds more emphatic, as if contrasting with someone else or stressing who should do it.

In ordinary instructions, Spanish usually leaves the subject out.

No, that would not work here.

Despacio describes how you should pet the cat, not how the cat should be afraid. So it belongs with acaricia, not with tenga miedo.

Natural:

• Acaricia al gato despacio para que no tenga miedo.

Not natural in this meaning:

• ...para que no tenga miedo despacio

That placement makes the adverb attach to the wrong idea.

Yes, acariciar is a very natural verb for stroking / petting / caressing.

Depending on context:

• acariciar = stroke, caress, pet
• tocar = touch
• mimar = pamper / spoil / fuss over

For an animal, acariciar is a good choice when you mean gentle contact with the hand.

So:

• Acaricia al gato despacio sounds natural and idiomatic.

Yes, but the nuance changes a little.

• para que no tenga miedo = so that it is not afraid / does not feel afraid
• para que no se asuste = so that it does not get frightened / startled

Both are natural, but they focus on slightly different things:

• tener miedo = the emotional state of fear
• asustarse = becoming scared, often as a reaction

So both are possible, but your original sentence is perfectly natural.