A szabadpolc az olvasóterem mellett van.

Both a and az mean the. Hungarian chooses between them based on the following sound:

• a before a consonant sound
• az before a vowel sound

So:

• szabadpolc starts with sz, which is a consonant sound
• olvasóterem starts with o, which is a vowel sound

It works a bit like English a/an.

Szabadpolc is a compound noun:

• szabad = free, open
• polc = shelf

In library language, it means something like open-access shelf or open stacks.

It is written as one word because Hungarian very often writes compound nouns as single words.

This is also a compound noun:

• olvasó = reader / reading
• terem = room, hall

So olvasóterem means reading room.

Hungarian uses compounds very productively, so learners will see a lot of words built this way.

Because mellett is a postposition.

English usually uses prepositions, which come before the noun: next to the reading room.

Hungarian often uses postpositions, which come after the noun phrase:

• az olvasóterem mellett

So the structure is literally something like the reading room beside.

Mellett usually means:

• beside
• next to
• at the side of

It normally suggests fairly close physical position.

If you want a looser idea like near, Hungarian more often uses words such as közel or közelében.

Because the location relationship is being shown by the separate word mellett, not by a suffix.

Hungarian can express location in two common ways:

• with a case ending
  
  • example: olvasóteremben = in the reading room
• with a postposition
  
  • example: olvasóterem mellett = beside the reading room

So in this sentence, olvasóterem stays in its basic form, and mellett carries the location meaning.

That is a very common question. Hungarian often omits the present-tense is in 3rd person singular, but not in every kind of sentence.

It is usually omitted in sentences like:

• A szoba nagy. = The room is big.
• Ő tanár. = He is a teacher.

But in location or existence sentences, van is normally kept:

• A könyv az asztalon van. = The book is on the table.

Your sentence is a location sentence, so van is needed.

That is a normal, neutral Hungarian word order.

A simple way to look at it is:

• A szabadpolc = topic, what the sentence is about
• az olvasóterem mellett = location information
• van = verb

Hungarian word order is flexible, but the order changes emphasis. For example:

• Az olvasóterem mellett van a szabadpolc.

This version puts more attention on the location.

Because both nouns are being treated as specific, identifiable things:

• a szabadpolc = the open-access shelf
• az olvasóterem = the reading room

Hungarian uses definite articles very naturally with known places and objects. In a sentence like this, having both articles sounds completely normal.

A few very useful pronunciation points:

• sz is pronounced like English s
• s is pronounced like English sh
• stress usually falls on the first syllable of each word

So:

• szabadpolc begins with an s sound
• olvasóterem contains s, so that part sounds like sh
• mellett has doubled consonants, which are pronounced a bit longer

A rough English-style guide:

• A szabadpolc az olvasóterem mellett van
• uh SAW-bawd-polts awz OL-vaw-shoh-teh-rem MEL-lett vawn

It does not match English perfectly, but it is a useful approximation for beginners.