Ayaw kong humiram.

Ayaw means not want, don't want, or sometimes refuse. It is the common opposite of gusto (want / like).

So:

• gusto ko = I want
• ayaw ko = I don't want

In this sentence, Ayaw kong humiram, ayaw expresses unwillingness: I don't want to borrow.

Because the -ng here is a linker.

• ko = my / me / I (in this kind of structure)
• ko + -ng = kong

The linker connects ko to the next word, humiram.

So:

• Ayaw ko = I don't want
• Ayaw kong humiram = I don't want to borrow

This -ng is not the same as the separate word ng used as a marker. Here it is just a linker attached to ko.

Because ko already carries that meaning in this structure.

In English, we say I don't want to borrow.
In Filipino, the pattern is more like:

• Ayaw ko = literally something like Not wanted by me / I don't want
• humiram = to borrow

So ko is the pronoun that tells you who feels the unwillingness: me / I.

This is normal in Filipino. You do not need a separate ako here.

Humiram means to borrow.

It comes from the root hiram, which is related to borrowing/loaning. The -um- is a verb marker inserted into the root:

• hiram → humiram

So humiram is the actor-focused verb form meaning borrow.

Here it means to borrow.

This can confuse learners because humiram can also be used in other contexts to mean borrowed. But after words like ayaw, it works like an infinitive in English:

• Ayaw kong humiram = I don't want to borrow

So in this sentence, do not read it as past tense.

It means borrow, not lend.

That is an important distinction:

• humiram = borrow
• magpahiram = lend

So Ayaw kong humiram means I don't want to borrow, not I don't want to lend.

Yes. Ayokong humiram is a very common contracted form.

It comes from:

• Ayaw ko → Ayoko
• Ayoko + -ng → Ayokong

So these mean the same thing:

• Ayaw kong humiram
• Ayokong humiram

The contracted form is especially common in everyday speech and informal writing.

The extra na changes the meaning.

• Ayaw kong humiram = I don't want to borrow
• Ayaw ko nang humiram = I don't want to borrow anymore

In nang, what you have here is basically na + -ng before the next word.

So if you see nang in this kind of sentence, it often adds the idea of already / anymore / now depending on context.

No. Just like in English, you can leave it unstated if it is already understood from context.

So Ayaw kong humiram is a complete sentence.

If you want to be more specific, you can add the thing being borrowed:

• Ayaw kong humiram ng pera. = I don't want to borrow money.
• Ayaw kong humiram ng libro. = I don't want to borrow a book.

You can also add the source:

• Ayaw kong humiram ng pera sa kanya. = I don't want to borrow money from him/her.

Yes. It is natural and very common.

Both of these are normal:

• Ayaw kong humiram
• Ayokong humiram

The first is slightly more full in form, while the second is the very common contracted version. Both sound natural.