L'asola è vicina al polsino.

Because Italian normally shortens la to l' before a word that begins with a vowel.

So:

• la asola → l'asola

This is called elision. The noun asola is still feminine singular, and its full article is la.

È with an accent is the verb is from essere.

Italian uses the accent here to distinguish it from e, which means and.

So:

• è = is
• e = and

This is a very important spelling difference.

Because vicina agrees with asola, which is a feminine singular noun.

Agreement here works like this:

• vicino = masculine singular
• vicina = feminine singular
• vicini = masculine plural
• vicine = feminine plural

Since l'asola is feminine singular, Italian uses vicina.

Al is a contraction of:

• a + il = al

In this sentence, vicino/vicina normally goes with a, so vicina al polsino means close to the cuff.

This kind of contraction is very common in Italian:

• a + il = al
• a + lo = allo
• a + la = alla
• a + l' = all'
• a + i = ai
• a + gli = agli
• a + le = alle

In Italian, vicino/vicina is commonly followed by a when you say what something is near.

So the pattern is:

• essere vicino/a a qualcosa = to be near something

Examples:

• La casa è vicina alla stazione.
• Il tavolo è vicino alla finestra.
• L'asola è vicina al polsino.

This is just the normal construction Italian uses.

Yes. Vicina is an adjective, not an adverb.

It describes the subject l'asola, and it appears after the verb è. That means it is being used as a predicate adjective.

So the structure is:

• L'asola = subject
• è = verb
• vicina = adjective describing the subject
• al polsino = complement telling you what it is near

Polsino is masculine singular.

That matters because it affects the article in al:

• a + il polsino = al polsino

But it does not control vicina.
Vicina agrees with l'asola, the subject, not with polsino.

In standard Italian, no.
It should be vicina, because asola is feminine.

So:

• L'asola è vicina al polsino. ✔
• L'asola è vicino al polsino. ✘

A learner may be tempted to leave vicino unchanged, but in Italian adjectives usually have to agree in gender and number.

Yes, a few alternatives are possible depending on style and focus.

For example:

• Il polsino è vicino all'asola.
  This shifts the subject to the cuff, but the basic relationship is the same.

• L'asola si trova vicino al polsino.
  This is a slightly more descriptive way to say the buttonhole is located near the cuff.

But the original sentence is completely natural and straightforward.

You would need to change the article, the verb, the adjective, and possibly the noun after a.

For example:

• Le asole sono vicine ai polsini.

Breakdown:

• l'asola → le asole
• è → sono
• vicina → vicine
• al → ai if polsino becomes plural polsini

This is a good example of how much agreement Italian uses.

A helpful approximate pronunciation is:

LAH-so-la eh vee-CHEE-na al pol-SEE-no

A few notes:

• asola has stress on the first syllable: A-so-la
• vicina has stress on -ci-: vi-CI-na
• polsino has stress on -si-: pol-SI-no
• ci before i sounds like chee

Also, in natural speech, l'asola links smoothly together.