Nemám ani péřový míček, ani plastový míček.

Because ani ... ani ... is the normal Czech pattern for neither ... nor ....

So:

• ani péřový míček, ani plastový míček = neither a feather shuttlecock/ball nor a plastic one

A single ani can also mean not even in other contexts, but here it is part of the paired structure ani ... ani ....

Czech normally uses ani ... ani ... with a negative verb.

So:

• Nemám ani A, ani B. = I have neither A nor B.

A positive verb here would not be natural. Compare:

• Mám A i B. = I have both A and B.
• Nemám ani A, ani B. = I have neither A nor B.

Yes, they are in the accusative. They just look the same as the nominative.

Why?

• míček is a masculine inanimate singular noun.
• For this noun type, nominative and accusative are usually identical.
• The adjectives péřový and plastový agree with míček, and their masculine inanimate accusative singular also looks the same.

So the grammar is:

• nominative: péřový míček
• accusative: péřový míček

Same form, different function.

Míček is a diminutive form of míč.

Very often, míček simply means:

• small ball
• or the normal word for certain sports balls/shuttlecocks

Czech uses diminutives much more freely than English, so míček does not always sound cute or childish. In many contexts, it is just the natural everyday word.

No. You can also say:

• Nemám ani péřový, ani plastový míček.

That is perfectly natural.

Repeating the noun:

• Nemám ani péřový míček, ani plastový míček.

makes the contrast extra clear and can sound a bit more explicit. Both versions are correct.

Because in Czech, paired conjunctions like ani ... ani ... are normally separated by a comma between the two coordinated parts.

So:

• ani péřový míček, ani plastový míček

is standard punctuation.

Yes. Czech word order is flexible.

For example:

• Nemám ani péřový míček, ani plastový míček. = neutral
• Ani péřový míček, ani plastový míček nemám. = more emphatic

The second version puts stronger focus on the things you do not have.

Sometimes, but it would not be the same structure.

• Nemám žádný míček. = I don’t have any ball/shuttlecock.

If you want to say neither ... nor ..., ani ... ani ... is the natural pattern:

• Nemám ani péřový míček, ani plastový míček.

You can also combine the ideas in a different sentence, for example:

• Nemám žádný míček — ani péřový, ani plastový.

A few useful points:

• Czech stress is usually on the first syllable
• é, í, ý are long vowels
• č sounds like ch in check
• ř is a special Czech sound with no exact English equivalent

Roughly:

• péřový → stress on pé-
• míček → stress on mí-

A very rough learner-friendly guide:

• míček ≈ MEE-chek
• péřový is harder because of ř; it is something like a rolled r mixed with a zh sound

The main thing is not to replace ř with plain English r forever, because it is a real Czech sound you will want to practice.