Караоке-бар открыт, а каталог песен здесь.

In Russian, the verb to be is usually omitted in the present tense.

So:

• Караоке-бар открыт = The karaoke bar is open
• каталог песен здесь = the song catalog is here

Literally, Russian often sounds like:

• karaoke-bar open
• catalog of songs here

This is completely normal.
Russian does use forms of быть in the past and future:

• Караоке-бар был открыт = The karaoke bar was open
• Караоке-бар будет открыт = The karaoke bar will be open

Открыт is the short form of the adjective/participle, and it is very commonly used to describe a state:

• бар открыт = the bar is open
• магазин закрыт = the shop is closed

Открытый is the full form, and it usually behaves more like a regular adjective modifying a noun:

• открытый бар = an open bar
• открытая дверь = an open door

So in this sentence, открыт is used because the meaning is the karaoke bar is in the state of being open.

Открыт is masculine singular, which tells you that караоке-бар is treated as a masculine singular noun.

Russian short forms agree with the noun:

• masculine: открыт
• feminine: открыта
• neuter: открыто
• plural: открыты

Examples:

• дверь открыта = the door is open
• окно открыто = the window is open
• магазины открыты = the shops are open

Since бар is masculine, the sentence uses открыт.

This is a common way in Russian to write some compound nouns, especially when one part specifies the type of something.

So караоке-бар means a karaoke bar as one combined concept.

You will see similar patterns in Russian, such as:

• интернет-магазин = online store
• кафе-бар = café-bar
• выставка-продажа = exhibition-sale

The main part here is бар, so the whole word behaves grammatically like бар.

After a noun like каталог meaning catalog, Russian usually puts the thing being cataloged in the genitive case.

So:

• каталог чего? = catalog of what?
• каталог песен = catalog of songs

Here песен is the genitive plural of песня.

Compare:

• каталог книг = catalog of books
• список гостей = list of guests
• чашка чая = cup of tea

So каталог песни would not fit the normal pattern here.

А often connects two ideas by contrasting them or by shifting attention from one thing to another.

In this sentence:

• Караоке-бар открыт, а каталог песен здесь.

it feels like:

• The karaoke bar is open, and/as for the song catalog, it’s here.

It is not a strong contradiction like but, but it does mark a change of focus.

Very roughly:

• и = simple and
• но = stronger but
• а = and / while / whereas / as for..., depending on context

So а is natural because the speaker is moving from one piece of information to another related but separate one.

Russian word order is flexible, and the end of the sentence often gets special emphasis.

So:

• каталог песен здесь

puts emphasis on здесь = here.

It answers the implied question Where is the song catalog?

You could also say:

• Каталог песен здесь.
• Здесь каталог песен.

Both are possible, but they sound a little different in focus.

• Каталог песен здесь = emphasis on here
• Здесь каталог песен = emphasis more on what is located here

By itself, здесь just means here.
But because Russian usually leaves out the present-tense is, the whole phrase:

• каталог песен здесь

means:

• the song catalog is here

So здесь does not itself mean is here; the idea of is is understood from the structure of the sentence.

In this sentence, открыт means open as a state, not opened as a completed action in the English sense.

So:

• бар открыт = the bar is open

Russian short-form participles/adjectives like this often describe the current condition of something.

Depending on context, forms from открыть can sometimes relate to the idea of something having been opened, but here the natural meaning is simply open for business.

Yes, especially the second part. Russian often allows very compact, natural phrasing.

• Караоке-бар открыт, а каталог песен здесь.

The second clause is basically:

• А каталог песен здесь. = And the song catalog is here.

This sounds natural in conversation, especially if someone is asking practical questions like where things are.

A more expanded version could be:

• Караоке-бар открыт, а каталог песен находится здесь.

But находится would usually sound more formal or heavier than necessary.

Песня means song.
Its genitive plural form is песен.

This is one of the forms you need after nouns like каталог:

• песня = song
• песни = songs / of the song, depending on case
• песен = of songs

So:

• каталог песен = catalog of songs

This is a good example of how Russian noun endings change depending on grammatical role, not just singular vs. plural.

Not in this sentence if you mean The karaoke bar is open.

Караоке-бар открытый sounds incomplete or unnatural here, because the full form открытый normally modifies a noun rather than serving as the usual predicate for this meaning.

Natural Russian says:

• Караоке-бар открыт.

If you use открытый, it would usually be in a phrase like:

• открытый караоке-бар
• открытый для всех бар = a bar open to everyone

So for the simple statement of state, открыт is the correct choice.