J'ai failli oublier mon portefeuille à la maison.

Here, faillir means to almost or to nearly do something.

The very common pattern is:

• faillir + infinitive

So:

• J'ai failli oublier... = I almost forgot...

A few similar examples:

• J'ai failli tomber. = I almost fell.
• On a failli rater le train. = We almost missed the train.
• Tu as failli me faire peur. = You almost scared me.

In modern everyday French, this is the main use learners see first.

J'ai failli is the passé composé of faillir.

It is built like this:

• j'ai = I have
• failli = past participle of faillir

So grammatically, it is:

• avoir + past participle

French uses the passé composé here because the near-miss happened at a specific moment in the past: you almost forgot your wallet.

Because faillir is followed directly by another verb in the infinitive.

This is normal in French with verbs that behave a bit like English to almost, to want to, to be able to, to have to, etc.

So the structure is:

• J'ai failli oublier
• not J'ai failli oublie
• not J'ai failli oublié

After failli, you use the base dictionary form of the next verb:

• failli partir
• failli tomber
• failli oublier

Yes. In French, oublier quelque chose quelque part can mean to forget something and leave it somewhere.

So:

• J'ai oublié mon portefeuille à la maison. means
• I forgot my wallet at home / I left my wallet at home by forgetting it.

That means oublier can cover both the mental idea of forgetting and the practical result of leaving something behind.

If you said:

• J'ai laissé mon portefeuille à la maison.

that simply means I left my wallet at home, but it does not automatically imply that it was accidental or due to forgetfulness.

Because French normally uses a possessive adjective here when talking about your personal belonging.

So:

• mon portefeuille = my wallet

This is the most natural choice in this sentence.

Using le portefeuille would usually sound less natural unless the context already makes it completely obvious whose wallet it is, or unless you are speaking in a more specific stylistic context.

For personal belongings, French often keeps the possessive:

• J'ai perdu mes clés.
• Elle a oublié son téléphone.
• Tu as pris ton sac ?

Here, à la maison means at home.

It is the normal idiomatic expression for at home in French.

So:

• à la maison = at home

In this sentence, it tells you where the wallet would have been forgotten or left behind.

Because à la maison is the idiomatic way to say at home, while dans la maison means inside the house.

Compare:

• Je suis à la maison. = I am at home.
• Je suis dans la maison. = I am inside the house.

The second one focuses on physical location inside the building. The first one is the everyday expression for being at home.

So in your sentence, à la maison is the natural choice.

Yes, that is understandable, and native speakers do use presque with verbs.

But faillir + infinitive is especially common and idiomatic for a near-miss action:

• J'ai failli oublier mon portefeuille.

It often sounds a bit more natural when you mean I nearly did this, but I didn’t.

So both can work, but faillir + infinitive is a very standard way to express almost did.

Because French commonly puts place expressions after the verb and its object.

So this order is very natural:

• J'ai failli oublier mon portefeuille à la maison.

The phrase à la maison goes with the action of forgetting/leaving the wallet behind. It tells you the place connected with that action.

A learner might wonder whether it means while I was at home. In practice, the natural reading is:

• the wallet would have been forgotten there
• the wallet would have been left at home

So the location is attached to mon portefeuille being forgotten, not mainly to where you were standing when you thought about it.

You can do it like this:

Question:

• Est-ce que j'ai failli oublier mon portefeuille à la maison ?
• Ai-je failli oublier mon portefeuille à la maison ?
  This second version is more formal.

Negative:

• Je n'ai pas failli oublier mon portefeuille à la maison.

The negative goes around ai:

• ne ... pas

So the pattern is:

• Je n'ai pas failli + infinitive

That is exactly how you would negate the sentence.