Nu, ni komencu la koncerton.

Because -u is the Esperanto verb ending for a wish, suggestion, request, or command.

So:

• ni komencas = we begin / we are beginning
• ni komencu = let’s begin

In this sentence, the speaker is not simply stating a fact. They are encouraging the group to do something, so -u is the right ending.

Yes. This is the standard Esperanto pattern for let’s ...:

• ni iru = let’s go
• ni manĝu = let’s eat
• ni atendu = let’s wait

So ni komencu literally means something like let us begin, but in natural English it is usually translated as let’s begin.

It is koncerton because it is the direct object of komenci.

In Esperanto, direct objects take the ending -n.

• la koncerto = the concert
• la koncerton = the concert (as the object of a verb)

Here, the thing being begun is the concert, so it gets -n:

• Ni komencu la koncerton. = Let’s begin the concert.

La is the definite article, meaning the.

So:

• koncerto = a concert / concert
• la koncerto = the concert

In this sentence, the speaker is referring to a specific concert that both speaker and listeners know about, so la is natural.

Nu is a very common little word in Esperanto. It often means something like:

• well
• so
• now then

It helps set the tone of what comes next. In this sentence, Nu makes the start sound a bit more natural and conversational:

• Nu, ni komencu la koncerton.
  = Well, let’s begin the concert.
  or So, let’s begin the concert.

It is not essential to the grammar; it is more of a discourse particle.

Yes, but it means something different.

• Ni komencu la koncerton. = Let’s begin the concert.
  (suggestion / invitation / urging the group)
• Ni komencas la koncerton. = We are beginning the concert / We begin the concert.
  (statement of fact)

So if someone is addressing a group and proposing action, komencu is the better choice.

Yes, you could. Esperanto word order is fairly flexible because the grammar endings show the roles of the words.

• Ni komencu la koncerton.
• La koncerton ni komencu.

Both are grammatical. The second version gives extra emphasis to la koncerton.

Still, Ni komencu la koncerton is the most neutral and natural order for most situations.

Yes. Komenci can be used transitively, so you can begin something directly:

• komenci la lecionon = begin the lesson
• komenci la ludon = begin the game
• komenci la koncerton = begin the concert

This is very normal Esperanto.

You can also use it without a direct object in some contexts:

• La koncerto komenciĝas. = The concert is beginning.

That version uses komenciĝi, which means to begin / to start in the sense of to start by itself or to get underway.

This is a very useful distinction.

• komenci = to begin/start something
• komenciĝi = to begin/start (intransitive: the subject itself begins)

Examples:

• Ni komencu la koncerton. = Let’s begin the concert.
  (we start it)
• La koncerto komenciĝas. = The concert is beginning.
  (the concert itself is starting)

So in your sentence, komenci is used because we are acting on the concert.

It is very normal to write a comma after Nu when it is used as a conversational opener.

• Nu, ni komencu la koncerton.

The comma reflects a slight pause, much like in English with Well, ... or So, ...

You may sometimes see punctuation vary in informal writing, but the comma is a good standard choice here.

Absolutely.

• Ni komencu la koncerton. = Let’s begin the concert.
• Nu, ni komencu la koncerton. = Well/So, let’s begin the concert.

Leaving out Nu makes the sentence a little more direct. Adding Nu makes it sound a bit more natural, conversational, or scene-setting.