Minél nedvesebb az agyag, annál könnyebb gyúrni.

It is a very common Hungarian pattern meaning the more ..., the ... or the -er ..., the -er ....

So:

• Minél nedvesebb az agyag, annál könnyebb gyúrni.
• literally: The wetter the clay, the easier to knead.

A good way to remember it:

• minél = introduces the changing condition
• annál = introduces the result that changes together with it

You will see this structure a lot, for example:

• Minél többet olvasol, annál jobban beszélsz.
  The more you read, the better you speak.

• Minél hidegebb van, annál több ruhát veszek fel.
  The colder it is, the more clothes I put on.

Because -bb is the normal Hungarian comparative ending, equivalent to English -er or more.

So:

• nedves = wet
• nedvesebb = wetter

and

• könnyű = easy, light
• könnyebb = easier, lighter

In this sentence:

• nedvesebb = wetter
• könnyebb = easier

So the sentence compares two things:

• how wet the clay is
• how easy it is to knead

Because here nedvesebb is part of the predicate, not directly modifying the noun inside a noun phrase.

Compare:

• a nedvesebb agyag = the wetter clay
  Here the adjective is attributive: it directly describes the noun.

• az agyag nedvesebb = the clay is wetter
  Here the adjective is predicative: it tells you something about the clay.

In your sentence, the structure is basically:

• Minél nedvesebb az agyag...
• The wetter the clay is...

So az agyag is the subject, and nedvesebb is the predicate adjective.

Hungarian often uses the definite article in places where English does not, especially with general or known things.

So az agyag here is perfectly natural even though English might just say clay.

This is not necessarily talking about one specific lump of clay only; it can also be a general statement about clay as a material. Hungarian is simply more comfortable using the article here.

So:

• az agyag = the clay / clay

depending on context.

Because Hungarian uses an adjective with an infinitive in this kind of structure, just like English uses easy to do.

So:

• könnyű / könnyebb + infinitive
• easy / easier + to do something

Examples:

• Könnyű megérteni. = It is easy to understand.
• Nehéz elhinni. = It is hard to believe.
• Könnyebb gyúrni. = It is easier to knead.

If you used könnyen, that is an adverb meaning easily, and it would usually go with a finite verb:

• Agyagot könnyen lehet gyúrni. = Clay can be kneaded easily.

So in your sentence, könnyebb is exactly the right form.

Because the sentence means something like it is easier to knead. Hungarian often uses an infinitive after words like easy, hard, possible, worthwhile, and so on.

So:

• gyúrni = to knead
• könnyebb gyúrni = easier to knead

This is an impersonal construction. English often uses a dummy subject like it:

• It is easier to knead.

Hungarian usually does not need an equivalent of that it.

Because the object is understood from the context: it is obviously the clay.

Hungarian often leaves out an object if it is clear already.

So:

• Minél nedvesebb az agyag, annál könnyebb gyúrni. = The wetter the clay, the easier it is to knead.

You could make the object explicit, for example:

• Minél nedvesebb az agyag, annál könnyebb gyúrni az agyagot.

But this sounds more repetitive and less natural unless you want special emphasis or extra clarity.

Gyúrni most basically means to knead or to work something by pressing and folding it with the hands.

It is commonly used with things like:

• dough
• clay
• soft materials shaped by hand

So with agyag it is a very natural verb.

Depending on context, gyúrni can also be used more broadly for working, molding, or manipulating a soft material by hand.

Because Hungarian word order here follows the normal predicate pattern:

• az agyag nedvesebb = the clay is wetter

In the minél ... annál ... structure, the comparative element often comes right after minél, so you get:

• minél nedvesebb az agyag

This puts focus on the thing that is changing: wetter.

So the order is not random. It helps highlight the comparison.

Yes, that is grammatical.

It means the same thing, but it is more explicit because it includes the object az agyagot.

Compare:

• annál könnyebb gyúrni = more natural, because the object is obvious
• annál könnyebb az agyagot gyúrni = more explicit, slightly heavier style

In everyday speech, Hungarian often prefers the shorter version when the meaning is already clear.

Yes. Hungarian often leaves out the verb to be in the present tense, third person, when it links a subject and an adjective or noun.

For example:

• Az agyag nedves. = The clay is wet.
• literally: The clay wet.

So in your sentence:

• Minél nedvesebb az agyag...
• the idea of is is understood

This is completely normal Hungarian grammar.