Bei der Moderation wertet sie alles aus und nennt unseren Teamnamen.

Here bei does not mean physical location only. With events or activities, bei often means during, as part of, or in the course of.

So bei der Moderation means something like:

• during the moderation
• while moderating
• as part of the hosting/presenting

A useful note: German Moderation is often used for hosting, presenting, or moderating a discussion/event. It is not just the English idea of moderation meaning self-control or not doing something too much.

Because bei always takes the dative case.

The noun is:

• nominative: die Moderation
• dative: der Moderation

So:

• bei die Moderation ❌
• bei der Moderation ✅

This is a very common pattern:

• bei dem Lehrer
• bei der Besprechung
• bei der Arbeit

This is because German main clauses follow the verb-second rule.

The first position is taken by Bei der Moderation. Once that happens, the finite verb must come next:

• Bei der Moderation | wertet | sie | alles | aus

So the subject sie comes after the verb.

Compare:

• Sie wertet alles aus.
• Bei der Moderation wertet sie alles aus.

Both are correct. The second one emphasizes the time/situation first.

It is one verb: auswerten.

This is a separable verb:

• infinitive: auswerten
• in the sentence: wertet ... aus

In a normal main clause, the prefix aus- moves to the end:

• Sie wertet alles aus.

In an infinitive or participle, it stays together:

• Sie muss alles auswerten.
• Sie hat alles ausgewertet.

auswerten usually means:

• to evaluate
• to analyze
• to process results/data

alles means everything here.

It is the direct object of auswerten, so it is in the accusative.

You can think of the structure like this:

• sie = subject
• wertet aus = verb
• alles = the thing being evaluated/analyzed

So:

• Sie wertet alles aus. = She evaluates/analyzes everything.

Because nennen takes an accusative object, and Teamname is masculine.

The basic noun is:

• nominative: der Teamname
• accusative: den Teamnamen

With the possessive adjective unser-, that becomes:

• nominative: unser Teamname
• accusative: unseren Teamnamen

So in this sentence, unseren Teamnamen is correct because it is the object of nennt.

Because Teamname behaves like Name, which is a masculine noun that changes form in some cases.

Compare:

• nominative: der Name
• accusative: den Namen

And likewise:

• nominative: der Teamname
• accusative: den Teamnamen

So both parts change in unseren Teamnamen:

• the adjective changes: unser → unseren
• the noun changes: Teamname → Teamnamen

This is something learners often notice because English nouns usually do not change like this.

nennt is from nennen, which can mean:

• to name
• to call
• to mention
• to state

In this sentence, the most natural meaning is probably:

• she says/announces our team name
• she names our team

So nennt unseren Teamnamen is not about giving the team a new name. It more likely means she states or announces the team name.

Yes, that is grammatically correct.

The difference is mainly emphasis:

• Bei der Moderation wertet sie alles aus und nennt unseren Teamnamen.
  This puts focus on the situation/time first.

• Sie wertet bei der Moderation alles aus und nennt unseren Teamnamen.
  This starts more neutrally with she.

German is flexible about what comes first, but the finite verb still stays in second position in a main clause.

It can be.

An English speaker may first think of moderation as:

• not too much of something
• self-restraint

But German Moderation very often means:

• moderating
• hosting
• presenting
• leading a discussion or event

So in this sentence, Moderation is about the role of a moderator or host, not the idea of restraint.