A lakásnézésen a bérbeadó négyzetméterben ellenőrzi az alapterületet.

A and az are the same definite article, like English the.
Which form you use depends on the sound that follows:

• a before a consonant: a lakásnézésen, a bérbeadó
• az before a vowel: az alapterületet

So az alapterületet is not a different grammar form from a bérbeadó; it is just the version used before a vowel sound.

Lakásnézésen is lakásnézés + -en.

Here -en is the superessive ending, usually taught as on / on top of, but in real Hungarian it is also very commonly used for events and occasions.

So:

• a lakásnézésen = at the apartment viewing
• not literally on the apartment viewing in English, even though that is the historical logic of the case

This is very common with event words:

• órán = in class / at class
• találkozón = at a meeting
• esküvőn = at a wedding

Because Hungarian normally uses -on/-en/-ön with events.

A learner often expects -ban/-ben because English says in, but Hungarian does not map cases to English prepositions one-to-one.

Compare the usual pattern:

• a koncerten = at the concert
• az órán = in class / at class
• a megbeszélésen = at the discussion/meeting

By contrast, -ban/-ben is more typical for being inside a place or in a state/category:

• a lakásban = in the apartment
• a szobában = in the room

So a lakásnézésen is the normal idiomatic form.

Bérbeadó means landlord / lessor, literally the one who rents it out.

It comes from:

• bér = rent
• bérbe ad = to rent out / to let
• adó = giving / one who gives

So bérbeadó is a built-up word meaning the person who gives something out for rent.

A useful contrast:

• bérbeadó = landlord, the one offering the property
• bérlő = tenant, the one renting it

Here -ben means in in the sense of in what unit something is expressed, measured, or checked.

So négyzetméterben means:

• in square meters
• measured in square meters
• using square meters as the unit

This is very common with units and currencies:

• forintban = in forints
• euróban = in euros
• literben = in liters
• kilóban = in kilos

So this is not a location meaning like inside square meters. It is a unit-of-measure use.

Because Hungarian has definite and indefinite verb conjugation.

Here the verb has a definite direct object:

• az alapterületet = the floor area

Since the object is specific and definite, Hungarian uses the definite verb form:

• ellenőrzi = he/she checks it

Compare:

• ellenőriz = he/she checks, is checking something non-specific
• ellenőrzi = he/she checks the specific thing

So in this sentence, ellenőrzi is the expected form.

That is the accusative ending, which marks the direct object.

The base word is:

• alapterület = floor area / base area

As the thing being checked, it becomes the object:

• alapterületet

The accusative marker is basically -t, but Hungarian often inserts a linking vowel, so you get forms like:

• könyv → könyvet
• nevet from név
• alapterület → alapterületet

So az alapterületet is the floor area as the object of the verb.

Because Hungarian forms compounds very freely, and many ideas that English writes as multiple words are written as one compound word in Hungarian.

For example:

• lakásnézés = lakás
  • nézés
    literally something like apartment-viewing
• alapterület = alap
  • terület
    literally base area
• négyzetméter = square meter
• bérbeadó is also a compact built form based on bérbe ad

This is very normal in Hungarian. A native English speaker often feels these words are long, but they are usually just several meaningful pieces joined together.

No. Hungarian word order is fairly flexible, and it often shows topic and focus rather than just subject-verb-object order.

In this sentence:

• A lakásnézésen sets the scene: at the apartment viewing
• a bérbeadó is the topic/subject: the landlord
• négyzetméterben is placed just before the verb, which gives it some prominence
• ellenőrzi az alapterületet = checks the floor area

So the sentence is grammatical as given, but other orders are possible if you want different emphasis.

For example:

• A bérbeadó a lakásnézésen ellenőrzi az alapterületet.
  More emphasis on the landlord as topic.
• Az alapterületet ellenőrzi a bérbeadó a lakásnézésen.
  More emphasis on the floor area.

So the order is not random, but it is not as rigid as in English either.

Yes: this sentence is a very good example of how Hungarian packs meaning into endings instead of separate prepositions.

You can see several different jobs being done by suffixes:

• lakásnézés-en → at the viewing
• négyzetméter-ben → in square meters
• alapterület-et → direct object
• ellenőr-z-i → verb form showing a definite object

So instead of several little helper words, Hungarian often attaches the grammar directly to the noun or verb. For English speakers, that is one of the biggest structural differences to get used to.