J’ai mis la dernière fiche dans mon casier, mais je crois que j’ai oublié mon stylo à la maison.

This is called elision. In French, je becomes j’ before a word that starts with a vowel sound or a silent h.

So:

• je ai → j’ai

French does this to make pronunciation smoother. You will see this often:

• j’aime
• j’habite
• j’oublie

J’ai mis is the passé composé, one of the main French past tenses.

It is formed with:

• the present tense of avoir or être
• plus a past participle

Here:

• ai = present of avoir
• mis = past participle of mettre

So j’ai mis literally looks like I have put, but in normal English it often translates as I put or I placed.

French uses the passé composé very often for completed past actions, especially in everyday speech.

Mis is the past participle of the verb mettre, which usually means to put, to place, or sometimes to set.

This verb is irregular, so the past participle is not something fully predictable. You need to learn it:

• mettre → mis

So:

• j’ai mis = I put / I placed

Because fiche is a feminine singular noun.

In French, articles and adjectives must agree with the noun in gender and number.

So:

• fiche = feminine singular
• article → la
• adjective → dernière

That gives:

• la dernière fiche

If the noun were masculine, you would have:

• le dernier ...

Good question. In this sentence, mis does not agree because it is used with avoir, and with avoir the past participle usually agrees only if the direct object comes before the verb.

Here, the direct object comes after the verb:

• J’ai mis la dernière fiche...

So mis stays unchanged.

If the object came before, agreement could happen:

• La fiche que j’ai mise...

The same idea applies to oublié in:

• j’ai oublié mon stylo

Because mon stylo comes after the verb, oublié does not change.

Dans means in or inside.

A casier is a locker, compartment, or cubby-like space, so French uses dans to show that something is placed inside it:

• dans mon casier = in my locker

If you said à mon casier, that would mean something more like at my locker or by my locker, not inside it.

French uses a possessive adjective here because the locker and the pen belong to the speaker.

• mon casier = my locker
• mon stylo = my pen

This works very much like English in this sentence.

Just note that French does not always match English in possession. For example, with body parts French often uses the definite article instead of a possessive. But for ordinary objects like casier and stylo, mon is completely normal.

Que introduces the next clause. It often corresponds to English that.

So:

• je crois que... = I think that... / I believe that...

In English, that is often omitted:

• I think I forgot my pen

But in French, que is normally kept:

• je crois que j’ai oublié...

After an affirmative expression like je crois que, French normally uses the indicative, because the speaker is presenting the idea as something believed to be true or likely true.

So:

• Je crois que j’ai oublié mon stylo.

That is normal.

The subjunctive is more likely after negative or doubtful forms, such as:

• Je ne crois pas que...
• Crois-tu que... sometimes depending on context

So in your sentence, j’ai oublié is the expected form.

Yes. À la maison is a very common French way to say at home or home in this kind of context.

So:

• j’ai oublié mon stylo à la maison = I forgot my pen at home

This is idiomatic. It does not mainly focus on the physical building.

Compare:

• à la maison = at home
• dans la maison = inside the house
• chez moi = at my place / at home

All can be correct in the right context, but à la maison is very natural here.

Because there are two separate past actions:

• J’ai mis la dernière fiche dans mon casier
• j’ai oublié mon stylo à la maison

Each clause needs its own verb form.

The word mais links the two ideas and means but:

• I put the last card in my locker, but I think I forgot my pen at home.

So even though the subject is still I, French still needs the second j’ai for the second action.