Mein Lineal liegt in der Mappe, und den Radiergummi finde ich erst in der Jackentasche.

Because finden takes a direct object in the accusative case.

• der Radiergummi is the dictionary form, the nominative
• after finden, it becomes den Radiergummi, the accusative

So:

• Der Radiergummi ist hier. = the eraser is here
• Ich finde den Radiergummi. = I find the eraser

In the first clause, Mein Lineal is the subject, so it stays nominative.

German main clauses follow the verb-second rule: the finite verb must be in the second position.

In den Radiergummi finde ich erst in der Jackentasche:

1. den Radiergummi is moved to the front for emphasis
2. the verb finde must still stay in second position
3. so the subject ich comes after the verb

This is called topicalization. English does something similar for emphasis sometimes, but much less often.

A more neutral version would be:

Ich finde den Radiergummi erst in der Jackentasche.

Yes. That is completely correct and sounds more neutral.

The original version,

Den Radiergummi finde ich erst in der Jackentasche,

puts extra focus on den Radiergummi. It helps create a contrast with the first clause:

• Mein Lineal liegt in der Mappe,
• but den Radiergummi I only find in the jacket pocket.

So both are grammatical, but the original has a stronger stylistic emphasis.

Because in is a two-way preposition. It can take:

• dative for location
• accusative for direction/movement into

Here both phrases describe where something is, not movement:

• Mein Lineal liegt in der Mappe. → location
• Ich finde den Radiergummi in der Jackentasche. → location

So German uses the dative:

• in der Mappe
• in der Jackentasche

If there were movement into something, you would use the accusative:

• Ich lege das Lineal in die Mappe.
• Ich stecke den Radiergummi in die Jackentasche.

Here erst means something like only or not until then/in that place.

So:

Den Radiergummi finde ich erst in der Jackentasche.

suggests that the speaker does not find the eraser at first, maybe after looking elsewhere, and then finally finds it in the jacket pocket.

Depending on context, English translations might be:

• I only find the eraser in the jacket pocket.
• I don’t find the eraser until I look in the jacket pocket.

So erst adds the idea of delay or only at that point.

German often prefers a position verb where English might simply use is.

liegen means to lie or to be lying. With objects, it is often used to say where something is located.

So:

• Mein Lineal liegt in der Mappe.

is more natural and more specific than just saying it is there.

German commonly uses verbs like:

• liegen = lie
• stehen = stand
• hängen = hang
• stecken = be stuck / be in

You could sometimes say ist, but liegt sounds more idiomatic here.

Because German forms compound nouns by joining nouns together.

• Jacke = jacket
• Tasche = pocket / bag

So Jackentasche literally means jacket-pocket, or more naturally, jacket pocket.

A very important rule is that the last part decides the gender and basic meaning of the compound. Since die Tasche is feminine, die Jackentasche is also feminine.

That is why in the sentence you get the dative form:

• in der Jackentasche

Because mein behaves a lot like ein, and in nominative singular neuter, it has no ending.

Lineal is neuter: das Lineal

So:

• nominative neuter: mein Lineal
• accusative neuter: mein Lineal
• dative neuter: meinem Lineal

Compare with other genders:

• masculine nominative: mein Radiergummi
• feminine nominative: meine Mappe

So the form mein here tells you that Lineal is singular, neuter, and nominative.

They are:

• das Lineal = ruler
• die Mappe = folder / file / school folder
• der Radiergummi = eraser
• die Jackentasche = jacket pocket

In the sentence, they appear in different case forms:

• Mein Lineal → nominative
• in der Mappe → dative
• den Radiergummi → accusative
• in der Jackentasche → dative

Knowing the dictionary gender helps you understand why the articles change the way they do.

Not strictly. With two main clauses joined by und, German often allows the comma to be omitted.

So both of these are possible:

• Mein Lineal liegt in der Mappe und den Radiergummi finde ich erst in der Jackentasche.
• Mein Lineal liegt in der Mappe, und den Radiergummi finde ich erst in der Jackentasche.

The comma is often used to make the structure clearer, especially when the clauses are a bit longer.

Yes. It gives that part extra emphasis and creates a contrast with Mein Lineal in the first clause.

The sentence is not just listing two facts. It is also subtly contrasting them:

• the ruler is in one place
• the eraser turns up only in another place

By fronting den Radiergummi, the speaker highlights that item in particular. It can sound a bit like:

As for the eraser, I only find it in the jacket pocket.

So the word order is not random; it helps shape the focus of the sentence.