Der Browser, mit dem ich arbeite, speichert mein Lesezeichen genau an der Stelle, an der der Cursor zuletzt war.

Because this is a relative clause referring back to der Browser.

• Main noun: der Browser
• Relative clause: mit dem ich arbeite

The pronoun dem stands for dem Browser and must match it in gender and number. Since mit always takes the dative, you get mit dem.

So the structure is:

• der Browser
• mit dem = with which / with whom
• ich arbeite = I work

A very literal version would be:

• The browser, with which I work, ...

In natural English we would usually say the browser I work with, but German normally keeps the preposition together with the relative pronoun: mit dem.

Because the preposition mit always requires the dative case.

So:

• mit dem Browser
• therefore in the relative clause: mit dem ich arbeite

It is not chosen because of the job the word has in English; it is chosen because German grammar requires dative after mit.

Quick comparison:

• der = nominative masculine / feminine dative / genitive feminine, depending on context
• den = accusative masculine
• dem = dative masculine or neuter

Here, Browser is masculine, singular, and after mit, so dem is the only correct form.

Not really a strong logical rule here—this is mostly a matter of grammatical gender.

In German, der Browser is simply the standard form. Many borrowed technical nouns ending in -er are masculine, for example:

• der Computer
• der Scanner
• der Router
• der Browser

So while you often just have to learn the gender with the noun, der Browser fits a common pattern.

Yes, Lesezeichen can be singular or plural. The form does not change.

You tell the difference from the article or possessive word:

• mein Lesezeichen = my bookmark (singular)
• meine Lesezeichen = my bookmarks (plural)

Here it is clearly singular because of mein.

Also, Lesezeichen is a neuter noun:

• das Lesezeichen
• mein Lesezeichen

Because an is a two-way preposition. That means it can take:

• dative for location
• accusative for direction/movement toward

Here the meaning is at that place/position, not to that place, so German uses the dative:

• an der Stelle = at the place / at the position

Compare:

• Ich bin an der Stelle. = I am at that place.
• Ich gehe an die Stelle. = I go to that place.

In your sentence, the browser stores the bookmark at the place where the cursor last was, so dative is correct.

Because the second part is another relative clause, and the preposition belongs there too.

Break it up like this:

• an der Stelle = at the place
• an der der Cursor zuletzt war = at which the cursor was most recently

The second an is not optional, because the relative clause needs the same preposition that belongs to Stelle in this meaning.

So German says, literally:

• at the place, at which the cursor last was

This is very normal German. English often avoids this kind of repetition more easily, but German keeps the preposition in both parts.

They are doing two completely different jobs.

1. an der
   Here der is a relative pronoun referring back to der Stelle.
   Since Stelle is feminine and an here takes the dative, it becomes an der.

2. der Cursor
   Here der is just the definite article for the noun Cursor.
   Cursor is masculine, so in the nominative it is der Cursor.

So although the words look identical, one is a relative pronoun and the other is an article.

Because both mit dem ich arbeite and an der der Cursor zuletzt war are subordinate clauses—more specifically, relative clauses.

In German subordinate clauses, the finite verb normally goes to the end.

So:

• ich arbeite becomes final in the clause mit dem ich arbeite
• der Cursor zuletzt war has war at the end

This is one of the most important German word order rules.

Zuletzt means last, most recently, or the last time depending on context.

Here:

• der Cursor zuletzt war = where the cursor was last / where the cursor most recently was

It sits before the verb because in a subordinate clause the verb goes to the end, and adverbs like zuletzt usually come somewhere before that final verb.

So the order:

• der Cursor
• zuletzt
• war

is completely normal German.

Because German uses commas to mark relative clauses very consistently.

In this sentence there are two of them:

1. mit dem ich arbeite
2. an der der Cursor zuletzt war

German punctuation is stricter than English here. If you see a relative pronoun such as der, die, das, dem, den starting a relative clause, a comma is usually required.

It is in the accusative case, because it is the direct object of speichert.

The verb is:

• speichern = to save / store

So:

• Der Browser ... speichert mein Lesezeichen ...

asks what does the browser store/save?
Answer: mein Lesezeichen

Since Lesezeichen is neuter, the form mein Lesezeichen looks the same in nominative and accusative, but grammatically it is accusative here.

Not with the same meaning.

• mit dem ich arbeite = with which I work
• bei dem ich arbeite usually means for which / at which I work

So:

• der Browser, mit dem ich arbeite = the browser I use / work with
• die Firma, bei der ich arbeite = the company I work for / where I work

For a tool or software you use, mit is the natural choice.

It is somewhat written or careful in style, but it is not unnatural.

German often uses relative clauses where English might prefer a simpler structure. A more conversational rewording might be possible, but the given sentence is perfectly normal, especially in written language.

What may feel formal to an English speaker is mainly:

• the use of two relative clauses
• the preposition + relative pronoun structure, such as mit dem and an der

That is standard German grammar, not unusually elevated language.