Ho comprato una molletta in più, perché ieri il vento ne ha portata via una.

Because this sentence uses the passato prossimo, the tense very commonly used in spoken Italian for a completed past action.

• ho comprato = I bought / I have bought
• The action is seen as finished.

Also, comprare takes avere as its auxiliary, so you say:

• ho comprato
• not sono comprato

That is because comprare is normally a transitive verb: you buy something.

In più means extra, additional, or one more.

So:

• una molletta in più = one extra clothespin / one more clothespin

In Italian, in più usually comes after the noun it modifies:

• un giorno in più = one extra day
• due euro in più = two extra euros
• una sedia in più = one extra chair

That word order is very natural in Italian. English often prefers an extra..., but Italian commonly uses noun + in più.

Here ne means something like of them or from them, and it refers back to the previously understood group of mollette.

In this sentence, it helps express:

• one of them
• meaning one clothespin out of the clothespins

A useful pattern is:

• ne ... uno/una/due/molti
  = one/two/many of them

For example:

• Ne ho comprate due. = I bought two of them.
• Ne manca una. = One of them is missing.

So in your sentence, ne avoids repeating molletta/e.

Because ne and una do different jobs.

• ne = of them
• una = one

Together, ne ... una means:

• one of them

So the structure is not redundant. It is a normal Italian pattern:

• Ne voglio una. = I want one of them.
• Ne ho viste tre. = I saw three of them.
• Il vento ne ha portata via una. = The wind carried one of them away.

Because the past participle can agree with the object being referred to, and here that object is understood as una molletta, which is feminine singular.

So:

• portata agrees with una molletta
• feminine singular noun -> feminine singular participle

This happens especially when a direct object pronoun comes before the verb, or with ne when the quantity is specified:

• Ne ho comprata una.
• Ne ha portata via una.

That said, in everyday spoken Italian, many speakers also say:

• ne ha portato via una

So portata is grammatically very neat and explicit, but portato is also commonly heard in less careful speech.

Not without changing the meaning.

• la = it, referring to one specific feminine object
• ne ... una = one of them, referring to a member of a group

So:

• Il vento l'ha portata via.
  = The wind carried it away.
  (one specific clothespin)

But:

• Il vento ne ha portata via una.
  = The wind carried one away / one of them away.
  (one from a set of clothespins)

So ne is the better choice here because the idea is one out of several.

Because portare via is the expression meaning to carry away / take away / blow away.

In compound tenses, via normally stays after the past participle:

• ha portato via
• ha portata via
• l'ha portata via

So:

• il vento ne ha portata via una

is the natural order.

You can think of portare via as a fixed verb expression, even though via is written separately.

Its position is fairly flexible.

The sentence has:

• perché ieri il vento ne ha portata via una

This is very natural, because ieri is placed early and sets the time right away.

But you could also hear other orders, such as:

• perché il vento ieri ne ha portata via una
• perché il vento ne ha portata via una ieri

The original version is probably the most neutral and smooth.

Yes. You could say:

• ... perché ieri il vento ha portato via una molletta.

That is perfectly correct.

But using ne is more elegant and natural because it avoids repetition:

• first mention: una molletta in più
• then refer back with ne ... una

Italian often prefers this kind of pronoun when the noun is already understood from context.