Наш репертуар пока небольшой, но в нём уже есть три старые песни.

In Russian, the verb to be is usually omitted in the present tense when you are simply linking a subject with a description.

So:

• Наш репертуар пока небольшой
  literally: Our repertoire for now small

This is normal Russian.
In the past or future, Russian does use forms of быть:

• Наш репертуар был небольшим — our repertoire was small
• Наш репертуар будет небольшим — our repertoire will be small

Here есть does not work like the English copula is. It means something more like:

• there is / there are
• it contains
• it has

So:

• в нём уже есть три старые песни
  = there are already three old songs in it

This is an existence/presence pattern, so есть is natural.

Compare:

• Репертуар небольшой — The repertoire is small.
  No есть.
• В репертуаре есть три песни — There are three songs in the repertoire.
  Есть is needed.

In this sentence, пока means for now, so far, or at the moment.

So Наш репертуар пока небольшой means:

• our repertoire is still small
• our repertoire is small for now

This use of пока often suggests the situation may change later.

It does not mean while here.

Небольшой is a very common single-word adjective meaning:

• small
• not very big
• modest

Writing it as one word is the normal choice here.

• небольшой репертуар = a small/modest repertoire

Writing не большой separately is usually used for contrast or special emphasis, something like:

• не большой, а маленький — not big, but small

So in your sentence, небольшой is the natural form.

Both can relate to small size, but they are not always equally natural.

• маленький often means little / small in size
• небольшой often means small, limited, modest in extent

With something abstract like репертуар, небольшой sounds more natural and idiomatic.

So:

• небольшой репертуар = a modest/small repertoire
• маленький репертуар is understandable, but less natural here

Because репертуар is a masculine singular noun in the nominative case.

Russian words that describe or determine a noun have to agree with it in gender, number, and case.

So:

• наш — masculine singular nominative
• небольшой — masculine singular nominative
• репертуар — masculine singular nominative

That is why the forms match.

Because the preposition в means in, and here it requires the prepositional case.

The pronoun referring back to репертуар is он.
After в, it becomes нём:

• он — he/it
• в нём — in him / in it

So:

• в нём = in it

Here it refers to репертуар.

A useful spelling note: in many texts, ё is often written as е, so you may also see в нем, but it is still pronounced в нём.

After the numerals 2, 3, 4, Russian normally uses the noun in the genitive singular.

The noun песня has:

• nominative singular: песня
• genitive singular: песни
• genitive plural: песен

So:

• три песни = correct
• три песен = incorrect

You would use песен after 5 and higher:

• пять песен — five songs

This is a very common learner question.

With feminine nouns after 2, 3, 4, the noun is still in the genitive singular:

• песни

But the adjective is commonly in the nominative plural:

• старые

So:

• три старые песни — three old songs

This is the normal standard pattern here.

A short way to remember it:

• numeral: три
• adjective: старые
• noun: песни

The noun form песни happens to look the same as nominative plural, but here it is functioning as genitive singular after три.

Уже means already.

It shows that even at this stage, something has been achieved or is already true.

So:

• в нём уже есть три старые песни
  = there are already three old songs in it

Without уже, the sentence would just state a fact.
With уже, it suggests progress.

Yes. Russian word order is fairly flexible, and changing it often changes emphasis, not the core meaning.

The sentence as given is neutral and natural:

• Наш репертуар пока небольшой, но в нём уже есть три старые песни.

Other orders are possible, for example:

• Но в нём уже есть три старые песни.
• Но уже в нём есть три старые песни.
• Но три старые песни в нём уже есть.

These versions sound more marked or put emphasis in different places.
For a learner, the original word order is the best neutral model to follow.