Deze fles azijn is moeilijk te openen als je handen nat zijn.

Because fles is a de-word in Dutch: de fles.

For singular nouns:

• deze is used with de-words
• dit is used with het-words

So:

• deze fles = this bottle
• dit boek = this book

In deze fles azijn, azijn functions like vinegar in this bottle of vinegar.

Dutch often leaves out van and also does not add a separate article in this kind of expression:

• een glas water = a glass of water
• een kop koffie = a cup of coffee
• een fles azijn = a bottle of vinegar

So azijn here is just the substance inside the bottle, not a separate noun phrase needing de or het.

Because Dutch usually does not use van in common container, quantity, and measure expressions.

Very common patterns are:

• een glas melk = a glass of milk
• een kom soep = a bowl of soup
• een fles azijn = a bottle of vinegar

Using van here would sound unnatural in normal Dutch.

This is a very common Dutch pattern:

adjective + te + infinitive

It often corresponds to English too/very ... to ... or, more often here, ... to ... after words like easy, hard, difficult.

Examples:

• makkelijk te begrijpen = easy to understand
• moeilijk te vinden = hard to find
• leuk om te lezen = nice/fun to read

So:

• moeilijk te openen = difficult to open / hard to open

It has a kind of passive meaning, yes, even though it is not a full passive form.

Deze fles azijn is moeilijk te openen literally looks like This bottle of vinegar is difficult to open, where the bottle is the thing receiving the action.

Dutch often uses this structure instead of a fuller passive like:

• Deze fles azijn kan moeilijk geopend worden
  = This bottle of vinegar can be opened only with difficulty / is hard to open

The version with moeilijk te openen is more natural and simpler.

Both are possible, but je is the normal, unstressed possessive form.

• je handen = your hands
• jouw handen = your hands, but with extra emphasis or contrast

For example:

• Was je handen. = Wash your hands.
• Niet mijn handen, maar jouw handen zijn nat. = Not my hands, but your hands are wet.

In this sentence, no special emphasis is needed, so je handen is the natural choice.

It actually does mean exactly your hands are wet.

The clause is:

• als = if/when
• je handen = your hands
• nat = wet
• zijn = are

The reason the verb comes last is that als introduces a subordinate clause, and in Dutch subordinate clauses the finite verb normally goes to the end.

Compare:

• Main clause: Je handen zijn nat. = Your hands are wet.
• Subordinate clause: als je handen nat zijn = if your hands are wet

Because nat is a predicate adjective here, not an adjective directly before a noun.

Compare:

Before a noun:

• natte handen = wet hands
  Here the adjective usually gets -e.

After a linking verb like zijn:

• de handen zijn nat = the hands are wet
  Here the adjective normally has no -e.

So:

• je handen nat zijn is correct
• je natte handen would also be correct, but that means your wet hands as a noun phrase

Because there are really two clauses, and each clause has its own verb.

1. Main clause:

   • Deze fles azijn is moeilijk te openen
   • The finite verb is is

2. Subordinate clause:

   • als je handen nat zijn
   • The finite verb is zijn

So the sentence contains:

• is for deze fles
• zijn for je handen

That is completely normal.

It can feel like either, depending on context.

In this sentence, als is best understood as:

• if your hands are wet or
• when your hands are wet

Dutch als often covers both kinds of meaning in everyday sentences like this.

If you want to think about the practical meaning, it is basically:

• the bottle is hard to open when/if your hands are wet

Yes, that is also good Dutch:

• Deze fles azijn is moeilijk te openen als je handen nat zijn.
• Deze fles azijn is moeilijk te openen als je natte handen hebt.

Both mean nearly the same thing.

A small difference in feel:

• je handen nat zijn focuses on the state of your hands
• natte handen hebt uses the common Dutch pattern have wet hands

Both are natural.

The sentence has a main clause followed by a subordinate clause.

Main clause:

• Deze fles azijn = subject
• is = finite verb
• moeilijk te openen = complement

Subordinate clause:

• als = subordinating conjunction
• je handen = subject
• nat = complement
• zijn = finite verb at the end

So the pattern is:

[main clause] + [als + subordinate clause]

This is very typical Dutch word order.

Not by itself in this sentence. Here deze is a demonstrative determiner directly attached to the noun:

• deze fles = this bottle

If you wanted this one, Dutch could use:

• deze by itself, if the noun is understood from context

For example:

• Welke fles bedoel je? = Which bottle do you mean?
• Deze. = This one.

But in your sentence, deze simply means this before fles.