Der Ärmel meiner Jacke ist nass geworden.

Because Ärmel is a masculine noun in the singular: der Ärmel = the sleeve.

A few forms are:

• der Ärmel = the sleeve
• die Ärmel = the sleeves

So in your sentence, it is talking about one sleeve, not both.

Because meiner Jacke is in the genitive case, which is often used to show possession or belonging.

So:

• der Ärmel meiner Jacke = the sleeve of my jacket / my jacket’s sleeve

Here is the structure:

• der Ärmel = the sleeve
• meiner Jacke = of my jacket

Since Jacke is feminine (die Jacke), the genitive form with mein becomes meiner Jacke.

Compare:

• meine Jacke = my jacket
• der Ärmel meiner Jacke = the sleeve of my jacket

Yes, you can, and native speakers do say that in everyday spoken German.

• der Ärmel meiner Jacke = more standard / more formal
• der Ärmel von meiner Jacke = common in speech

Both are understandable. The version with the genitive (meiner Jacke) is especially useful to learn because it is very common in written and careful German.

This is the present perfect of werden.

• werden can mean to become
• nass werden = to get wet / to become wet

So:

• ist ... geworden = has become / got

In this sentence:

• Der Ärmel meiner Jacke ist nass geworden.
• Literally: The sleeve of my jacket has become wet.
• Natural English: My jacket sleeve got wet.

Because werden forms its perfect tense with sein, not haben.

Many German verbs that show:

• a change of state,
• or movement from one place to another,

often use sein in the perfect tense.

Examples:

• Er ist müde geworden. = He got tired.
• Das Kind ist eingeschlafen. = The child fell asleep.
• Ich bin nach Hause gegangen. = I went home.

So:

• Der Ärmel ... ist nass geworden is correct
• hat nass geworden is not correct

Because the full past participle of werden in this use is geworden.

• ist nass geworden = has gotten wet

You may also see worden, but that is usually in the passive, not in the become meaning.

Compare:

• Er ist krank geworden. = He became ill.
• Das Fenster ist geöffnet worden. = The window was opened.

So in your sentence, since werden means become, geworden is the correct form.

Because nass is being used as a predicate adjective, not directly before a noun.

After verbs like:

• sein = to be
• werden = to become
• bleiben = to stay

German adjectives usually do not take endings.

So:

• Der Ärmel ist nass.
• Der Ärmel ist nass geworden.

But when the adjective comes before a noun, it does take an ending:

• ein nasser Ärmel = a wet sleeve
• der nasse Ärmel = the wet sleeve

It means the sleeve became wet or got wet.

This suggests a change:

• before: it was not wet
• then: something happened
• now: it is wet

That is slightly different from just saying:

• Der Ärmel ist nass. = The sleeve is wet.

That sentence only describes the current state.
But:

• Der Ärmel ist nass geworden. = The sleeve got wet.

That emphasizes that it ended up wet.

The basic structure is:

• Der Ärmel meiner Jacke = subject
• ist ... geworden = verb phrase
• nass = adjective complement

German often puts the conjugated verb in second position, and the past participle at the end in a perfect-tense sentence.

So the pattern is:

• Subject + auxiliary verb + other information + past participle

Here:

• Der Ärmel meiner Jacke = subject
• ist = auxiliary verb
• nass = complement
• geworden = past participle at the end

No, not in standard German.

English can put two nouns together very easily, like my jacket sleeve, but German usually does not do that in the same way.

Instead, German uses structures like:

• der Ärmel meiner Jacke
• der Ärmel von meiner Jacke

So Meine Jacke Ärmel sounds ungrammatical.

Because meiner is an inflected form of the possessive word mein.

The base forms are:

• mein = my
• dein = your
• sein = his/its
• ihr = her/their
• etc.

But these words change form depending on:

• gender
• case
• number

Since Jacke is feminine and here it is in the genitive singular, mein becomes meiner:

• die Jacke = the jacket
• meine Jacke = my jacket
• der Ärmel meiner Jacke = the sleeve of my jacket

Yes, it is completely natural.

It sounds like a normal way to say that one sleeve of a jacket got wet. A native speaker might also say things like:

• Ein Ärmel meiner Jacke ist nass geworden. = One sleeve of my jacket got wet.
• Der Ärmel von meiner Jacke ist nass geworden.
• Mein Jackenärmel ist nass geworden. = My jacket sleeve got wet.

All of these are possible, but Der Ärmel meiner Jacke ist nass geworden. is perfectly correct and natural.