Non lasciare andare il cane in giardino senza guinzaglio.

Because this is the negative informal command for tu.

In Italian, when you tell one person not to do something, you usually use:

non + infinitive

So:

Non lasciare = Don’t let / Don’t allow

Compare:

• Lascia andare il cane. = Let the dog go. / Allow the dog to go.
• Non lasciare andare il cane. = Don’t let the dog go.

If you were speaking formally to Lei, you would say:

• Non lasci andare il cane...

So non lasciare is correct here because it is an informal negative command.

Lasciare andare is a very common verb combination in Italian.

Literally, it is:

• lasciare = to let, to allow, to leave
• andare = to go

Together, lasciare andare means:

• to let go
• to allow to go
• in this sentence, to let the dog go into the garden

So the structure is:

lasciare + infinitive

This is similar to English:

• Let him go
• Make him wait
• Help her carry it

Italian often uses this pattern too:

• Lascia andare il cane = Let the dog go
• Fammi parlare = Let me speak
• Aiutami a capire = Help me understand

Here, andare stays in the infinitive because it depends on lasciare.

Italian usually uses the definite article more often than English.

So where English may say:

• Don’t let the dog into the garden without a leash

Italian naturally says:

• Non lasciare andare il cane...

The article il is used because we are talking about a specific dog, or at least a clearly identifiable one in the situation.

In Italian, leaving the article out would usually sound wrong here.

Both can be possible, but they are not exactly the same in feel.

• in giardino often sounds a bit more general or idiomatic: into/in the garden
• nel giardino is more explicitly in the garden = in the specific garden

In many everyday contexts, Italian often says in giardino where English learners might expect nel giardino.

So:

• andare in giardino = to go into the garden / to go to the garden area

If you say nel giardino, you are highlighting the garden more as a specific physical place.

In this sentence, in giardino sounds very natural.

Because senza guinzaglio means without a leash in a general sense.

After senza, Italian often leaves out the article when speaking generally about something:

• senza zucchero = without sugar
• senza problemi = without problems
• senza scarpe = without shoes
• senza guinzaglio = without a leash

If you say senza il guinzaglio, that would usually mean without the leash, referring to a specific leash already known in the context.

So:

• senza guinzaglio = no leash on, generally
• senza il guinzaglio = without the specific leash

In practical terms, it describes the situation in which the dog is being allowed into the garden: the dog is without a leash.

So the meaning is:

• Don’t let the dog go into the garden while it is not on a leash

It is attached to the whole action, but semantically it tells us the dog’s condition during that action.

English works similarly:

• Don’t let the dog into the garden without a leash

That does not mean the garden has no leash; it means the dog is not leashed.

Yes, depending on context, lasciare andare can suggest letting go, releasing, or allowing to go.

In this sentence, the idea is probably:

• don’t allow the dog to go into the garden or
• don’t let the dog loose into the garden

So there is a slight overlap between allowing and releasing.

That is normal. Italian often leaves this kind of nuance to context.

Not with the same meaning.

• portare il cane in giardino = to take/bring the dog into the garden
• lasciare andare il cane in giardino = to let the dog go into the garden

So portare means you actively bring the dog there. lasciare andare means you allow the dog to go there.

Those are different actions.

It mainly suggests movement into the garden.

That is because of:

• lasciare andare = to let go / let go somewhere
• andare naturally implies movement

So andare in giardino here is best understood as:

• to go into the garden
• to go out into the garden

Even though in can sometimes mean simply in, with andare the idea is usually movement toward that place.

Yes, a little.

The given sentence is perfectly natural:

• Non lasciare andare il cane in giardino senza guinzaglio.

But Italian can move parts around for emphasis. For example:

• Non lasciare il cane andare in giardino senza guinzaglio.

This is understandable, but it is generally less smooth than lasciare andare kept together.

Since lasciare andare functions as a tight verb combination, keeping those two verbs together is usually the most natural choice.

You could say:

• Non lasciarlo andare in giardino senza guinzaglio.

This means:

• Don’t let him/it go into the garden without a leash.

Here:

• lo = him/it
• lasciarlo = to let him/it

With the negative informal command, both of these are possible in many cases:

• Non lo lasciare andare...
• Non lasciarlo andare...

Both are correct. The attached form lasciarlo is very common and natural.