Learn Hardcore German: Während der Klassenarbeit sitzt die Schülerin ganz konzentriert am Tisch. - During the class test the student sits fully focused at the table.

As a conjunction (introducing a subordinate clause), während means while and is followed by a full clause with the verb at the end:
- Während die Klassenarbeit läuft, sitzt die Schülerin …
  (läuft is at the end of the subordinate clause.)
In your sentence, während is a preposition meaning during. Prepositions are followed by a noun phrase, not a full clause:
- während der Klassenarbeit = during the test / during the class test

So you need a case-marked noun phrase, and not die Klassenarbeit as a subject of a verb.

Why is it der Klassenarbeit after während? What case is that?

Here der Klassenarbeit is genitive singular, because the preposition während normally governs the genitive.

Nominative: die Klassenarbeit (the test)
Genitive: der Klassenarbeit (of the test / during the test)

So:

während der Klassenarbeit = during the test (literally “during of-the-test”)

In everyday spoken German, many people use the dative after während, but for feminine nouns, genitive and dative look the same (der Klassenarbeit), so you can read it as “correct” genitive and not worry too much.

Why is the verb sitzt in second position, even though the sentence begins with Während der Klassenarbeit?

German main clauses follow the V2 rule: the finite verb must be in second position. Important: “position” means logical slot, not “second word”.

The phrase Während der Klassenarbeit is one single prepositional phrase, so it occupies position 1.
The finite verb sitzt must then come in position 2.
The rest (die Schülerin ganz konzentriert am Tisch) follows after the verb.

So the structure is:

Während der Klassenarbeit (element 1)
sitzt (verb in 2nd position)
die Schülerin ganz konzentriert am Tisch (rest of the clause)

If während were used as a conjunction introducing a subordinate clause, the verb of that clause would go to the end:

Während die Klassenarbeit läuft, sitzt die Schülerin …
(läuft is at the end of the subordinate clause.)

Could I also say Die Schülerin sitzt während der Klassenarbeit ganz konzentriert am Tisch? Is there any difference?

Yes, that sentence is perfectly correct and has (almost) the same meaning.

Die Schülerin sitzt während der Klassenarbeit ganz konzentriert am Tisch.
Während der Klassenarbeit sitzt die Schülerin ganz konzentriert am Tisch.

Both follow the V2 rule. The difference is emphasis / focus:

Starting with Die Schülerin: neutral, subject-first word order; you’re focusing on who is doing the action.
Starting with Während der Klassenarbeit: you foreground the time frame (“during the test”), making the situation or contrast more prominent (e.g., “During the test she is very focused, but normally she is not.”)

Why is it sitzt and not something like ist (as in “is sitting”)?

German doesn’t have a grammatical progressive form like English is sitting. The simple present can express both:

Sie sitzt am Tisch. = She sits at the table / She is sitting at the table.

Using sitzt instead of ist here is also semantically important:

Die Schülerin sitzt ganz konzentriert am Tisch.
→ Describes her posture (she is sitting) and her mental state (very focused).
Die Schülerin ist ganz konzentriert.
→ Only describes her mental state (“She is very focused”), but tells you nothing about her posture or location.

So sitzt … am Tisch paints a more concrete picture: she is physically sitting at the table, and in that posture she is mentally focused.

What exactly does ganz konzentriert mean here, and what is the role of ganz?

konzentriert is an adjective meaning concentrated / focused.
ganz here is an intensifier, similar to very, fully, or really.

So ganz konzentriert ≈ very focused / fully concentrated.

Note that ganz can have two main meanings in German:

whole / entire (attributive, before a noun):
- der ganze Tag = the whole day
very / completely / really (intensifier, before adjectives/adverbs):
- ganz konzentriert = very focused
- ganz leise = very quiet

In your sentence it is clearly the intensifier use, not “the whole concentrated”.

Grammatically, what is ganz konzentriert doing in the sentence? Why no adjective ending?

ganz konzentriert is a predicative adjective phrase that describes the subject’s state while she is sitting.

Subject: die Schülerin
Verb: sitzt
Predicative complement: (ganz) konzentriert

In German, adjectives only take endings when they are directly in front of a noun (attributive use):

die konzentrierte Schülerin (adjective before noun → needs ending)
die sehr konzentrierte Schülerin

But in predicative position (after verbs like sein, werden, bleiben, sitzen, stehen, liegen etc.), no ending is used:

Die Schülerin ist konzentriert.
Die Schülerin sitzt ganz konzentriert am Tisch.

So ganz konzentriert here functions like an adverbial description of how she is sitting.

Why is it am Tisch and not an dem Tisch or auf dem Tisch?

am is the contracted form of an dem:
- an dem Tisch → am Tisch

German very often contracts an dem to am in everyday language, especially in fixed expressions.

As for an vs auf:

an (here: am Tisch) ≈ at / by a vertical or defined surface/edge:
- am Tisch sitzen = sit at the table
auf dem Tisch = on the table (physically on top of the table)

So:

Die Schülerin sitzt am Tisch.
→ Normal collocation: she sits at the table on a chair.
Die Schülerin sitzt auf dem Tisch.
→ She is literally sitting on top of the table, which is unusual in a classroom.

Therefore am Tisch is the natural phrase here.

Why is it die Schülerin and not der Schüler? What does the ending -in mean?

Schülerin is the feminine form of Schüler.

der Schüler = male pupil / student (school-age), or sometimes gender-neutral in older usage
die Schülerin = female pupil / student

The ending -in is a common way to form feminine nouns for people:

der Lehrer → die Lehrerin (teacher)
der Student → die Studentin (university student)
der Kollege → die Kollegin (colleague)

In your sentence, die Schülerin is:

feminine, singular, nominative (because it is the subject of the verb sitzt).

What tense is sitzt? How would I say “was sitting” in German?

sitzt is present tense (Präsens).

German Präsens can express:

English simple present:
- Die Schülerin sitzt am Tisch. = The pupil sits at the table.
English present progressive:
- Die Schülerin sitzt am Tisch. = The pupil is sitting at the table.

To say “was sitting”, you normally use Präteritum (simple past) of sitzen:

Die Schülerin saß während der Klassenarbeit ganz konzentriert am Tisch.
= The pupil was sitting very focused at the table during the test.