When you face one of these issues, here is the "how to do it".

Problem:

I need to cancel the high bid and give the item to the next highest bidder.

Solution:

Go to Components > Auction > Online/Mobile tab.  Search for the item, click it to open.  You will see all of the bids received.  Cancel/Delete the highest bid and the system will automatically make the next highest bidder the winner.  An email will go out to that user saying something like "You won after all!"  The item will be in their checkout cart to purchase and will show up in the Financials > Pending table.

Problem:

A user wants a refund and I'd like to give the item to the next highest bidder.

Solution:

When an item is already paid for, it is no longer available.  A refund will not make it available again.  However, you can open the item in Components > Auction > Catalog.  Search the item, click to open, and inside the details page change the quantity to "2".  This creates another one of those items as available.  Change the begin and end times that it is available and also change the sale type to fixed price and set the price at the sale price. 

Then, follow the instructions above to locate the next highest bidder and reach out to advise them that they can purchase the item at their last bid.  Provide them with a link to purchase the item or create their bid for them in Online/Mobile.  At the end time, the bidder will win.