Der Mitspieler schüttelt den Würfelbecher vor der Spielrunde.

Because der Mitspieler is the subject of the sentence — the person doing the action.

• der Mitspieler = nominative masculine singular
• den Mitspieler = accusative masculine singular

In this sentence, the fellow player is the one shaking something, so German uses the nominative case:

• Der Mitspieler schüttelt ...

Because den Würfelbecher is the direct object — the thing being shaken.

The verb schütteln usually takes an accusative object, so masculine nouns change like this:

• nominative: der Würfelbecher
• accusative: den Würfelbecher

So:

• Der Mitspieler = the subject
• den Würfelbecher = the object

Mitspieler literally means co-player or fellow player.

It is made from:

• mit = with
• Spieler = player

Depending on context, it can mean:

• a fellow participant in a game
• another player at the table
• sometimes a teammate, but not always

So it does not automatically mean someone on your side. It often just means another player involved in the game.

schüttelt is the 3rd person singular present tense of schütteln, which means to shake.

Conjugation:

• ich schüttle
• du schüttelst
• er/sie/es schüttelt
• wir schütteln
• ihr schüttelt
• sie/Sie schütteln

Since der Mitspieler is equivalent to he, the correct form is:

• Der Mitspieler schüttelt ...

Because German main clauses normally follow the V2 rule: the finite verb goes in the second position.

Here the structure is:

1. Der Mitspieler
2. schüttelt
3. den Würfelbecher
4. vor der Spielrunde

So even if you moved something else to the front, the verb would still stay in second position:

• Vor der Spielrunde schüttelt der Mitspieler den Würfelbecher.

Because vor is a two-way preposition, and here it is used in a time sense: before the round of play.

With temporal expressions like this, German uses the dative:

• vor der Spielrunde

Also, Spielrunde is feminine:

• nominative: die Spielrunde
• dative: der Spielrunde

So:

• vor der Spielrunde = before the game round

Here vor takes the dative.

vor can take:

• accusative when there is movement toward a position
• dative when it describes a location or is used temporally

In this sentence, it means before in time, not movement in space, so it must be dative:

• vor der Spielrunde

Spielrunde is a compound noun:

• Spiel = game
• Runde = round

So Spielrunde means game round, round of play, or round depending on context.

Because Runde is feminine, the whole compound is feminine too:

• die Spielrunde

In German compounds, the last part determines the gender.

It is another compound noun:

• Würfel = dice
• Becher = cup / tumbler

So Würfelbecher literally means dice cup, the cup used for shaking and rolling dice.

Again, the final part controls the gender:

• der Becher is masculine
• therefore der Würfelbecher is also masculine

Yes. German word order is flexible, especially for time expressions.

All of these are possible:

• Der Mitspieler schüttelt den Würfelbecher vor der Spielrunde.
• Vor der Spielrunde schüttelt der Mitspieler den Würfelbecher.

Both are grammatical. The second version puts more emphasis on when it happens.

What stays the same in a normal main clause is that the finite verb stays in second position.

A few pronunciation points are especially useful here:

• sch is pronounced like sh in ship
• ü is a front rounded vowel that does not exist in normal English
• w in German sounds like English v
• ch in Becher is a soft sound, not like English ch in chair

Rough guides:

• schüttelt ≈ SHU-telt, but with German ü, not English u
• Würfelbecher ≈ VYUR-fel-bekh-er, again with ü

If ü is difficult, a common learning trick is:

1. Say ee
2. Keep your tongue in that position
3. Round your lips

That gets you closer to the German ü.

In this sentence, Mitspieler is singular:

• der Mitspieler = the fellow player

The plural is:

• die Mitspieler = the fellow players

So:

• singular: Der Mitspieler schüttelt den Würfelbecher.
• plural: Die Mitspieler schütteln den Würfelbecher.

Notice that in the plural, the verb also changes:

• singular: schüttelt
• plural: schütteln

Yes, if you mean a fellow player rather than the fellow player.

Compare:

• Der Mitspieler schüttelt den Würfelbecher. = The fellow player shakes the dice cup.
• Ein Mitspieler schüttelt den Würfelbecher. = A fellow player shakes the dice cup.

So the choice depends on whether the person is specific or not.

Because the meaning here is before the round, and German expresses that with vor + dative:

• vor der Spielrunde

By contrast:

• in der Spielrunde = in the round
• während der Spielrunde = during the round
• nach der Spielrunde = after the round

So the preposition changes according to the meaning you want.