Trotzdem legt meine Freundin den Taschenrechner neben das Arbeitsblatt.

Trotzdem means nevertheless, even so, or despite that. It connects this sentence to something said before and shows contrast.

So the idea is something like:
Something else happened or was expected, but even so, she puts the calculator next to the worksheet.

It is an adverb, not a conjunction like weil or dass.

German main clauses follow the verb-second rule. That means the finite verb must come in the second position.

Here, Trotzdem takes the first position, so the verb legt must come next:

• Trotzdem | legt | meine Freundin ...

Even though meine Freundin is the subject, it comes after the verb because Trotzdem is in position 1.

If you started with the subject instead, you could say:

• Meine Freundin legt trotzdem den Taschenrechner neben das Arbeitsblatt.

That is also correct, but the emphasis is slightly different.

Because the subject is meine Freundin, which is third person singular: she.

The verb legen is conjugated like this in the present tense:

• ich lege
• du legst
• er/sie/es legt
• wir legen
• ihr legt
• sie/Sie legen

So with meine Freundin, you need legt.

Because Freundin is a feminine noun, and here it is the subject of the sentence, so it is in the nominative case.

The possessive word mein- changes its ending depending on gender, number, and case.

In the nominative singular:

• mein Bruder
• meine Freundin
• mein Kind

So meine Freundin is correct because Freundin is feminine.

Because den Taschenrechner is the direct object of legt — it is the thing being placed.

The noun Taschenrechner is masculine: der Taschenrechner.

Masculine nouns change in the accusative case:

• nominative: der Taschenrechner
• accusative: den Taschenrechner

So:

• meine Freundin = subject
• den Taschenrechner = direct object

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

• accusative for direction / movement toward a position
• dative for location / existing position

Here the verb is legen = to lay / put, which involves moving something into a new place. So German uses the accusative:

• neben das Arbeitsblatt = to a position next to the worksheet

Compare:

• Sie legt den Taschenrechner neben das Arbeitsblatt.
  She puts the calculator next to the worksheet.
  → movement, so accusative

• Der Taschenrechner liegt neben dem Arbeitsblatt.
  The calculator is lying next to the worksheet.
  → location, so dative

Also note that Arbeitsblatt is neuter:

• nominative/accusative: das Arbeitsblatt
• dative: dem Arbeitsblatt

German often chooses different verbs depending on the physical position of the object.

• legen = to lay something down, so it ends up lying
• stellen = to put/set something so it ends up standing
• setzen = to set someone/something so it ends up sitting

A calculator is normally thought of as being placed flat on a surface, so legen is the natural choice.

Related pair:

• legen = to lay something down
• liegen = to be lying

So:

• Sie legt den Taschenrechner ... = She lays/puts the calculator ...
• Der Taschenrechner liegt ... = The calculator is lying ...

Yes, absolutely.

Both are grammatical:

• Trotzdem legt meine Freundin den Taschenrechner neben das Arbeitsblatt.
• Meine Freundin legt trotzdem den Taschenrechner neben das Arbeitsblatt.

The difference is mainly emphasis:

• starting with Trotzdem highlights the contrast more strongly
• starting with Meine Freundin makes the subject more neutral or more prominent

German word order is often flexible, but the finite verb still stays in second position in a main clause.

The nouns are:

• die Freundin — feminine
• der Taschenrechner — masculine
• das Arbeitsblatt — neuter

You usually need to learn noun gender together with the article.

A few hints:

• nouns ending in -in for female people are feminine:
  die Freundin, die Lehrerin, die Studentin
• compound nouns take the gender of the last part:
  • der Taschenrechner because der Rechner is masculine
  • das Arbeitsblatt because das Blatt is neuter

So in compounds, the final element is very important for gender.

Yes. In this sentence, Trotzdem is capitalized because it starts the sentence.

Normally, trotzdem is written with a lowercase t when it appears in the middle of a sentence:

• Meine Freundin legt trotzdem den Taschenrechner neben das Arbeitsblatt.

German always capitalizes nouns, but trotzdem is not a noun here — it is an adverb.