Question: During a *pidyon haben* (redemption of 1st born son) the father gives the Cohen 5 silver shekels (or equivalent value.)

I'm not sure how much 5 silver shekel is worth in today's U.S. funds. But silver values are rising. I assume the Cohen can keep this money and do what he wishes, or must he use it only for holy purposes? What do Cohanim typically do with the money? Are they allowed to and do they ever return the money to the father?

Answer: 
The answer is from my own experience as a bechor. :) Or at least what i've been told about it...

When i was born (<20 years ago) my parents used 5 silver dollar coins for the pidyon, in accordance with common practice (and i suppose what their rabbi said).  

The kohen returned the money to us as a gift later (which my parents then gifted to me). I am told that this is actually pretty common. From here we can see that the money doesn't have to be used for holy purposes, and can be (and is!) returned to the father afterwards. Note that SA Yoreh Deah 305:8 rules that while it is permissible to return the money, it's not recommended.
At my son's Pidyon HaBen, I went to a coin store and bought 10 silver half dollars. The kohen kept them.
I was then in the Chabura learning Yoreh Deah in Bais Medrash Govoha. The Kohen was Rav Shmuel Meir Katz, even then was involved in p'sak in the Yeshiva. Now he is an accepted poseik for decades. 

As for the legal status of the money, it's clear from the fact that the kohen is allowed to give the money ...as a gift... to a poor man, there is no kedushah (sanctity) imparted to the coins. The whole hetter (permission) to return a poor man's money is to encourage the poor man to do a rather expensive mitzvah that he might feel unable to do. Rather than push it off, Chazal permitted the Kohen to let him do the mitzvah and then, free-willingly,the Kohen returns the money so that the poor man can use it.

As for a kohen who has a set of five silver dollars to sell to the father and then the father uses them for the Pidyon (see above answer), the idea is to use five coins at the Pidyon, as the simple reading of the Torah prescribes. As pointed out (by Double AA), though, the bottom line is coins must be used that have intrinsic value of approximate 117 grams of silver. Five U.S. silver dollars (or ten US half dollars) satisfy that. The idea of having coins always available and selling them to the father helps because those silver coins haven't been minted for almost 50 years and it becomes increasingly difficult to find them. So if the Kohen has five ready for use, it makes it easier to do the mitzvah. But still, the coins have no kedushah.