La doccetta e la guarnizione del gruppo sono vicine.

Because both doccetta and guarnizione are singular feminine nouns.

In Italian, the definite article must agree with the noun:

• la = singular feminine
• il / lo = singular masculine
• le = plural feminine
• i / gli = plural masculine

So:

• la doccetta
• la guarnizione

Even though the two nouns are joined by e (and), each noun still keeps its own article here.

Because the subject is two things:

• la doccetta
• la guarnizione del gruppo

When two singular nouns are joined by e, they make a plural subject, so the verb must also be plural.

That is why Italian uses:

• è = is for one thing
• sono = are for more than one thing

So:

• La doccetta è vicina. = The shower screen is close.
• La doccetta e la guarnizione del gruppo sono vicine. = The shower screen and the group gasket are close.

Because adjectives in Italian must agree with the noun or nouns they describe in gender and number.

Here, the two nouns are:

• la doccetta = feminine singular
• la guarnizione = feminine singular

Together, they form a feminine plural subject, so the adjective must also be feminine plural:

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

So sono vicine is the correct form.

Yes, in standard Italian, vicini would be wrong in this sentence.

Since both nouns are feminine, the adjective should be feminine plural: vicine.

Use vicini when:

• the group is all masculine, or
• it is a mixed group with at least one masculine noun

For example:

• Il filtro e il gruppo sono vicini.
• La doccetta e il filtro sono vicini.

But with two feminine nouns:

• La doccetta e la guarnizione sono vicine.

Del is a contraction of di + il:

• di = of
• il = the
• del = of the

So la guarnizione del gruppo literally means the gasket of the group.

Grammatically, del gruppo is attached to guarnizione and tells you which gasket it is.

So the structure is:

• La doccetta
• e
• la guarnizione del gruppo
• sono vicine

In other words, the phrase del gruppo modifies guarnizione, not doccetta.

Because gruppo is a masculine noun in Italian, and its normal singular article is il:

• il gruppo
• del gruppo = di + il gruppo

This is a good reminder that noun gender in Italian is grammatical, not logical. A word may be masculine or feminine simply because that is how it behaves in the language.

Here:

• guarnizione is feminine
• gruppo is masculine

So you get:

• la guarnizione
• del gruppo

Because this is a very common Italian pattern:

• subject + essere
  • adjective

So:

• La doccetta è vicina.
• Le parti sono vicine.

In this sentence, vicine is a predicate adjective: it comes after sono and describes the subject.

This works much like English are close.

Yes. Doccetta is built from doccia with the suffix -etta, which often gives a diminutive sense.

So doccetta can suggest something like:

• a small shower
• a little shower piece
• in technical contexts, a specific small shower-like component

This is very common in Italian. Diminutive suffixes include:

• -ino / -ina
• -etto / -etta
• -ello / -ella

Even when the exact technical meaning is already known, it is useful to notice that doccetta is morphologically related to doccia.

Normally, no. In standard Italian, that would sound wrong here.

When you mention two separate countable nouns joined by e, each noun usually keeps its own article:

• la doccetta e la guarnizione del gruppo

Leaving out the second article would not be natural in this sentence.

In this sentence, yes, it most naturally refers to physical proximity: the two parts are near each other.

The adjective vicino / vicina / vicini / vicine can sometimes also be used more abstractly, depending on context, but here the straightforward reading is spatial.

So sono vicine means that the two components are located near one another.