A járda a híd mellett van, a híd után egy kis tér van.

A is the Hungarian definite article, meaning the.

So:

• a járda = the sidewalk
• a híd = the bridge

Hungarian does not have grammatical gender, so the same article is used for masculine, feminine, and neuter nouns.

You will also see az, which is the form used before a vowel sound:

• a híd = the bridge
• az ablak = the window

Egy means a/an.

So:

• egy kis tér = a small square

It is used here because the square is being mentioned as a new, indefinite thing, not as a specific previously known square.

Compare:

• a tér = the square
• egy tér = a square

Because mellett is a postposition, not a preposition.

English usually puts these words before the noun:

• next to the bridge
• after the bridge

Hungarian often puts them after the noun:

• a híd mellett = next to the bridge
• a híd után = after the bridge

So the structure is normal Hungarian word order for this kind of expression.

Because postpositions like mellett and után normally follow the noun in its basic form.

So:

• a híd mellett
• a híd után

The noun does not take an extra case ending here.

This is different from forms where Hungarian uses a case ending instead of a postposition, for example:

• a hídon = on the bridge
• a hídnál = at/by the bridge

So in your sentence, mellett and után do the job themselves.

Mellett means beside, next to, or by the side of.

In this sentence:

• A járda a híd mellett van. = The sidewalk is next to the bridge.

It describes position very naturally.

Related forms you may also meet:

• mellé = to beside / to the side of
• mellől = from beside

Here után means after in a spatial sense:

• a híd után = after the bridge / past the bridge

It can also be used in a time sense:

• ebéd után = after lunch
• munka után = after work

So után is a very common word with both spatial and temporal uses.

Van is the 3rd person singular form of to be, meaning is.

Hungarian often puts van at or near the end in simple location or existence statements:

• A járda a híd mellett van.
• A híd után egy kis tér van.

This sounds natural and neutral.

The sentence patterns are basically:

• X Y mellett van = X is next to Y
• Y után Z van = After Y, there is Z

Yes.

Hungarian uses the same verb, van, in both kinds of sentences:

• A járda a híd mellett van. = The sidewalk is next to the bridge.
• A híd után egy kis tér van. = After the bridge there is a small square.

So English uses two different-looking structures, but Hungarian uses van in both.

Because Hungarian adjective order is:

• article + adjective + noun

So:

• egy kis tér = a small square
• a kis tér = the small square

Here:

• egy = a
• kis = small
• tér = square

That order is the normal one.

Both kis and kicsi mean small, but kis is especially common before a noun.

So:

• egy kis tér sounds very natural
• egy kicsi tér is possible, but less standard in this kind of neutral description

A useful rule of thumb:

• kis is very common as an attributive adjective before a noun
• kicsi is very common as a predicate or standalone adjective

For example:

• Ez a tér kicsi. = This square is small.

Yes. Hungarian word order is flexible, and changing it can shift the emphasis.

For example:

• A járda a híd mellett van. = neutral: The sidewalk is next to the bridge.
• A híd mellett van a járda. = more like It is next to the bridge that the sidewalk is

And in the second clause:

• A híd után egy kis tér van. = emphasizes what comes after the bridge
• Egy kis tér van a híd után. = emphasizes that there is a small square

Both can be grammatical, but the original sentence sounds like a natural descriptive route/location statement.

Hungarian can connect short related clauses with just a comma, especially in simple descriptive statements.

So this is natural:

• A járda a híd mellett van, a híd után egy kis tér van.

You could also say:

• A járda a híd mellett van, és a híd után egy kis tér van.

Adding és means and, but it is not necessary here.

A rough pronunciation guide is:

• A járda ≈ aw YAAR-daw
• a híd mellett ≈ aw heed MEL-lett
• van ≈ vawn
• a híd után ≈ aw heed oo-TAAN
• egy kis tér ≈ edj kish tayr

A few useful pronunciation points:

• Stress is usually on the first syllable of each word.
• Long vowels matter:
  • í in híd
  • á in járda
  • é in tér
• gy in egy is a single Hungarian sound, not an English g
  • y sequence.

In the third person singular present tense, Hungarian often omits the verb to be when you are simply saying X is Y:

• A híd nagy. = The bridge is big.

But when you are talking about location or existence, van is normally used:

• A híd nagy. = The bridge is big.
• A híd ott van. = The bridge is there.
• A járda a híd mellett van. = The sidewalk is next to the bridge.

So in your sentence, van is needed.