En la biblioteca hay veintidós mesas, pero hoy casi todas están ocupadas.

Because hay is used to say that something exists: there is / there are.

• Hay veintidós mesas = There are twenty-two tables
• It introduces the existence of the tables in the library.

By contrast, estar is used more for location or state/condition:

• Las mesas están en la biblioteca = The tables are in the library
• Las mesas están ocupadas = The tables are occupied

So in this sentence:

• hay veintidós mesas = existence
• están ocupadas = condition/state

Spanish often puts the place first to set the scene.

• En la biblioteca hay veintidós mesas
• Literally: In the library there are twenty-two tables

This is very natural in Spanish. You could also say:

• Hay veintidós mesas en la biblioteca

Both are correct. Starting with En la biblioteca just gives the location first, which often sounds smooth and natural.

In modern Spanish, many numbers from 21 to 29 are written as one word:

• veintiuno
• veintidós
• veintitrés
• veinticuatro, etc.

So veintidós is the standard spelling for 22.

It has a written accent because that is the standard spelling of the number 22.

Some numbers in the veinti- group carry an accent:

• veintidós
• veintitrés
• veintiséis

The accent shows the correct stress. In veintidós, the stress falls on the last syllable: veinti-DÓS.

Because todas refers back to mesas, and mesas is:

• feminine
• plural

So the word that replaces it must also be feminine plural:

• mesas → todas

That gives:

• casi todas = almost all of them

Compare:

• todos for masculine plural nouns
• todas for feminine plural nouns
• todo usually means everything or all in a singular/general sense

Yes. The full idea is:

• casi todas las mesas están ocupadas

But Spanish often leaves out the noun when it is already obvious from context. Since mesas was just mentioned, todas on its own is enough.

So:

• casi todas están ocupadas means
• almost all of them are occupied

Because estar is used for a state or condition, and here occupied describes the current situation of the tables.

• están ocupadas = they are occupied right now

Using ser here would sound wrong in normal speech. In many cases, ser + past participle suggests a passive construction focused on the action:

• Las mesas son ocupadas por los estudiantes = The tables are occupied by the students

That is grammatical, but much less natural here. The sentence is talking about the present condition of the tables, so estar is the right choice.

Because it agrees with mesas (or the implied mesas after todas).

Mesas is:

• feminine
• plural

So the adjective must also be:

• feminine
• plural

That gives:

• ocupada = feminine singular
• ocupadas = feminine plural

Compare:

• La mesa está ocupada
• Las mesas están ocupadas

Because todas still refers to mesas, which is plural.

So even though the noun is omitted, the grammar still follows the hidden plural idea:

• (las mesas) están ocupadas

That is why both the verb and adjective are plural:

• están
• ocupadas

Casi means almost.

So:

• todas = all (of them)
• casi todas = almost all (of them)

It modifies todas, not the verb. The idea is that not every table is occupied, but nearly every one is.

Hoy means today, and it gives a time reference for the second clause:

• pero hoy casi todas están ocupadas = but today almost all are occupied

It contrasts the general fact in the first clause with the specific situation today.

Its position is flexible. These are all possible, though some are more natural than others:

• pero hoy casi todas están ocupadas
• pero casi todas están ocupadas hoy

Putting hoy early often sounds more natural when you want to highlight the time contrast.

Pero means but and introduces a contrast.

The contrast is:

• In general, the library has twenty-two tables.
• Today, almost all of them are occupied.

So pero links those two ideas in a natural way.

Yes, absolutely. That version is also correct and natural.

The difference is mostly emphasis:

• En la biblioteca hay veintidós mesas puts the location first.
• Hay veintidós mesas en la biblioteca puts the existence of the tables first.

Both mean the same thing.

In standard Spanish, hay is pronounced like eye in English, more or less.

A simple guide:

• hay ≈ eye

It is the same form for:

• there is
• there are

So:

• hay una mesa = there is one table
• hay veintidós mesas = there are twenty-two tables

Biblioteca is stressed on the syllable te:

• bi-blio-TE-ca

So it sounds like:

• bee-blyo-TEH-ka

That stress follows normal Spanish pronunciation rules, so it does not need a written accent mark.

Usually mesa means table. In a library context, English might sometimes use desk depending on the kind of furniture, but mesas is the normal Spanish word here.

If the sentence specifically meant school desks, context would usually make that clear. In a library, tables is the safest translation.