Der Mitspieler beginnt die Spielrunde.

German nouns have grammatical gender, and the article has to match both the noun’s gender and its case.

• der Mitspieler: Mitspieler is masculine, and here it is in the nominative case because it is the subject.
• die Spielrunde: Spielrunde is feminine, and here it is in the accusative case because it is the direct object.

A useful detail: for feminine nouns, die is the same in both nominative and accusative singular, so you do not see a change there.

The subject is der Mitspieler, and the object is die Spielrunde.

You can tell this from:

• the verb meaning: someone begins something
• the case marking: der Mitspieler is nominative, which strongly suggests subject
• normal German word order: subject + verb + object is the most neutral pattern here

So:

• der Mitspieler = the one doing the action
• die Spielrunde = the thing being begun

Because the subject is third-person singular: der Mitspieler = he / the fellow player.

The infinitive is beginnen. In the present tense, it changes like this:

• ich beginne
• du beginnst
• er/sie/es beginnt
• wir beginnen
• ihr beginnt
• sie/Sie beginnen

So with der Mitspieler, you need beginnt.

In a normal German main clause, the finite verb usually comes in second position. This is often called the V2 rule.

In this sentence:

• Der Mitspieler = first position
• beginnt = second position
• die Spielrunde = the rest of the clause

So the sentence follows the standard German main-clause pattern perfectly.

Because beginnen can take a direct object in German.

So German says, structurally, something like:

• someone begins something

That means die Spielrunde can appear directly in the accusative, without a preposition.

This is different from some verbs that require a preposition, but beginnen does not need one here.

Mitspieler is a compound noun:

• mit- = with / fellow / co-
• Spieler = player

So Mitspieler literally means something like fellow player, co-player, or sometimes teammate, depending on context.

German uses compound nouns very often, and learning to spot the parts can make vocabulary much easier.

Spielrunde is also a compound noun:

• Spiel = game
• Runde = round

So Spielrunde literally means game round or round of play.

Again, this is very typical German: instead of using a longer phrase, German often combines nouns into one compound word.

Grammatically, Mitspieler is the masculine form.

Depending on context, it can mean:

• a specifically male fellow player
• sometimes a mixed or unspecified person in more traditional usage

If you want to be more explicit, you might also see:

• Mitspielerin = female fellow player
• Mitspielerinnen und Mitspieler = female and male fellow players
• Mitspielende = a more gender-neutral form

So yes, the form in your sentence is grammatically masculine.

Yes, but the basic sentence you have is the most neutral one.

For example, German can move elements to the first position for emphasis:

• Die Spielrunde beginnt der Mitspieler.

This is grammatically possible, but it sounds much more marked and context-dependent. The original sentence:

• Der Mitspieler beginnt die Spielrunde. is the straightforward, neutral version.

A key rule still stays the same: the finite verb beginnt remains in second position.

Because in German, all nouns are capitalized.

So:

• Mitspieler is a noun
• Spielrunde is a noun

That is why both begin with capital letters. This is one of the most noticeable spelling differences between German and English.

It is partly regular in the present tense, but overall it is considered a strong verb because its past forms change.

Present:

• ich beginne
• er beginnt

Past:

• ich begann

Past participle:

• begonnen

So in your sentence, the present-tense form looks fairly predictable, but the verb is not fully regular across all tenses.

Yes, very often.

German commonly uses both:

• beginnen
• anfangen

So a similar sentence could be:

• Der Mitspieler fängt die Spielrunde an.

That is very natural in everyday German. The difference is mainly style:

• beginnen can sound a bit more formal or neutral
• anfangen is often a bit more conversational

Also note that anfangen is a separable verb, so in a main clause it splits:

• fängt ... an