A kés a tányér mellett van, a villa pedig a kanál mellett.

Because Hungarian normally keeps van in location sentences.

Here the sentence says where something is:

• A kés a tányér mellett van.

That is different from simple 3rd-person present sentences of identity or description, where van is often omitted:

• A kés éles. = The knife is sharp.
• A villa tiszta. = The fork is clean.

So a good rule is:

• where something is → usually van
• what something is like / what something is → often no van in 3rd person present

Mellett means next to, beside, or by.

It comes after the noun because it is a postposition. English uses prepositions, which go before the noun, but Hungarian often uses postpositions, which go after it:

• a tányér mellett = next to the plate
• a kanál mellett = next to the spoon

So the order is naturally:

• noun phrase + postposition

Because mellett itself expresses the relationship.

In a tányér mellett and a kanál mellett, the nouns stay in their basic dictionary form, and mellett carries the meaning next to.

Hungarian can express place in two common ways:

• with a case suffix
  
  • a tányéron = on the plate
• with a postposition
  
  • a tányér mellett = next to the plate

So in this sentence, no extra case ending is needed on tányér or kanál.

It is omitted because it is already understood from the first clause.

The full version would be:

• A kés a tányér mellett van, a villa pedig a kanál mellett van.

That is grammatical, but Hungarian often leaves out repeated words when the meaning is clear. So the shorter version sounds more natural and less repetitive.

Here pedig adds a mild contrast or parallel idea. It is something like:

• and as for the fork...
• while the fork...
• a slightly contrastive and

So the sentence sets up a neat comparison:

• A kés ...
• a villa pedig ...

As for its position: pedig often comes after the topic or first important element of the clause. That is why Hungarian says:

• a villa pedig

rather than usually starting with pedig here.

Yes, you could say:

• A kés a tányér mellett van, és a villa a kanál mellett.

That is understandable and grammatical.

But pedig is often more natural when two parallel facts are being compared or balanced against each other. It gives the sentence a clearer this one..., and the other one... feeling.

So:

• és = plain and
• pedig = and / while / whereas, with a stronger sense of parallel contrast

Because each noun phrase has its own definite article:

• a kés
• a tányér
• a villa
• a kanál

Hungarian normally repeats the article for each separate noun phrase, just as English says:

• the knife
• the plate
• the fork
• the spoon

So the repetition is completely normal.

Hungarian has two forms of the definite article:

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

All four nouns here begin with consonants:

• kés
• tányér
• villa
• kanál

So the correct article is a each time.

No. Hungarian word order is flexible, but this sentence has a very natural, neutral structure.

In the first clause:

• A kés is the topic
• a tányér mellett van is what is said about it

So it works like:

• As for the knife, it is next to the plate.

If you change the word order, the emphasis changes. For example:

• A tányér mellett van a kés.

This puts more focus on the location, almost like answering:

• Where is the knife?

So the given word order is not the only grammatical one, but it is a standard, neutral choice.

In Hungarian, these accents mainly show vowel length, not stress.

So:

• é is a long vowel
• á is a long vowel

Examples in this sentence:

• kés has long é
• tányér has long á and long é
• kanál has long á

Also, Hungarian stress is usually on the first syllable of the word.

One more useful pronunciation point:

• ny in tányér sounds like the ny in canyon

So the accents are important: they are part of the spelling and pronunciation, not optional marks.