Al is a contraction of:
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:
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:
In standard Italian, no.
It should be vicina, because asola is feminine.
So:
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.