Answering in reverse order:
Suppose the holder of the prepaid warrant decides not to exercise on the expiry date. Will the warrant holder get a refund of the prepayment?
No, they will not get their money back. Both Prepaid Warrant – Definition from The Business Professor website (my emphasis):
However, owning a warrant doesn’t mean one already owns stock and therefore will receive dividends or will have a voting right. It is up to the investor to exercise the right, but not the obligation, bestowed by the warrant in exchange for real stock that can receive dividends from the company. If the warrant is not exercised before expiration date it becomes null and void and the purchase price will not be returned.
and Practice Pointers on Pre-funded Warrants [PDF] from the website of law firm Morrison Foerster:
However, a pre-funded warrant that expires unexercised has no further value and the purchase price of the pre-funded warrant paid by the holder at the time of the pre-funded warrant’s issuance is not returned to the holder upon expiration.
agree. That you do not get your money back immediately begs your first question:
What are prepaid warrants for?
At least one reason, according to the above PDF from Morrison Foerster, is to facilitate existing investors who might otherwise be barred from investing further due to limits on the proportion of shares owned:
Why Use Pre-funded Warrants?
The purpose of using pre-funded warrants is to provide investors that have restrictions on their ability to own a company’s securities above a designated ownership threshold (typically, 9.99% or 19.99%) with the opportunity to invest additional capital without violating the investor’s ownership restrictions. A pre-funded warrant provides a holder with the flexibility to avoid exceeding the designated ownership threshold prior to the warrant’s exercise while still maintaining the ability to immediately acquire the underlying securities at a nominal exercise price when the investor is ready to do so.
Other parts of that PDF imply that pre-funded warrants or not for the "causal investor", specifically:
Pre-funded warrants are generally issued as part of a larger financing transaction, such as a venture capital investment, minority equity investment, or mezzanine financing. The pre-funded warrant holder typically also invests in other types or classes of the company’s securities, including its common stock, convertible securities, and traditional warrants, as part of or concurrent with the holder’s acquisition of the pre-funded warrants.
Both The Business Professor (my emphasis):
Warrants can be traced back to the early 1920s in the United State. Many large corporations had stock warrants trading, for instance, AT&T, Bank of America, General Motors, Ford Motors and hundreds of others. However, presently Warrants are no longer popular in the United States but are heavily traded in Canada, Hong Kong, Germany and other several countries.
and Warrant on Investopedia:
Warrants are no longer common in the United States but are heavily traded in Hong Kong, Germany, and other countries.
note that the use of warrant (pre-paid or not) is no longer common within the US.