The buyer isn't necessarily lying to you when they say they didn't use PP.

These days, buyers are offered a few different ways to pay and as far as I know, the seller has not a clue how they paid as that information isn't shown to them. 

The payment goes straight to ebay, not straight to the seller.

That's what managed payment is all about.

Then, if you as seller want your money, you have to arrange it so ebay transfers the money across.

As a new seller, there would be a 30 hold on funds.

Why on earth you had a buyer who paid cash after already paying, I do not know. They should have checked beforehand that the money had been refunded. But it is what it is. It might only be $5 but for the buyer, it may the principle of the thing.

Even when a person is picking up an item, you are not safe to just take the money. These days, ebay has a special pick up scan that should be done. It proves that the person has in fact taken the item. That way, you shouldn't end up with a dishonest buyer claiming they never got it.

You say you've sold a few items, presumably mostly for cash.

You'll need to get into the help pages & get yourself properly verified for managed payments. ASAP.

That way you will eventually have access to the money. Plus ebay can get its fees.

Try to refund through ebay  (following countess's guide) and see how you go. That needs to be done immediately.

And don't list anything else on ebay till you sort yourself out.