Pecten avi nusquam inveniri potest.

Because this sentence uses a passive infinitive: inveniri means to be found, not to find.

In an active sentence, pectinem would be the direct object:

• Avis pectinem invenire potest = The bird can find the comb

But here the thing being found becomes the grammatical subject:

• Pecten ... inveniri potest = The comb ... can be found

So pecten is nominative singular.

It is a very common Latin pattern:

• potest = is able / can
• inveniri = present passive infinitive of invenire, to be found

Together:

• inveniri potest = can be found

This works much like English can + verb, except Latin uses possum plus an infinitive.

Avi is best understood here as dative singular of avis.

So it means something like:

• for the bird
• to the bird
• or a dative of reference/advantage, depending on context

A native English speaker often expects of the bird, but that would normally be avis, not avi.

Because of the bird would usually require the genitive case, and the genitive singular of avis is avis.

So:

• avis = of the bird
• avi = to/for the bird or another dative use

That is why avi should not normally be read as possession here.

Formally, yes: avi can be either dative singular or ablative singular.

But in this sentence, dative is the more natural interpretation, because there is no obvious ablative function such as:

• means/instrument
• separation
• place where
• accompaniment with cum

So although the form is ambiguous by itself, the syntax points more naturally to the dative.

Because its grammatical subject is pecten, which is singular.

Latin verbs agree with their subject in number:

• pecten ... potest = the comb ... can
• pectines ... possunt = the combs ... can

Even though avi is also singular, it is not the subject.

Nusquam is an adverb meaning nowhere.

So:

• nusquam inveniri potest = can be found nowhere

Latin does not need an extra non here.
In fact, nusquam already contains the negative idea.

English often prefers either:

• can be found nowhere or
• cannot be found anywhere

Latin uses nusquam for that negative location idea.

Latin word order is much freer than English word order because the endings show the grammatical relationships.

So even if the words move around, the cases still tell you what is happening:

• pecten = subject
• avi = dative
• nusquam = adverb
• inveniri potest = verbal idea

This means several orders are possible without changing the basic meaning. The given order may simply sound natural or emphasize nusquam.

Because Latin has no articles.

So pecten can mean:

• a comb
• the comb

Which one is intended depends on context, not on a separate word.

That is very normal in Latin and is something English speakers have to get used to.

A more active version could be something like:

• Avis pectinem nusquam invenire potest

That would mean The bird can find the comb nowhere or The bird cannot find the comb anywhere.

This is not necessarily identical in nuance to the original, but it helps show the grammar:

• active: pectinem as object
• passive: pecten as subject