Yes, Harry's name really IS Henry (Henry Charles Albert David, to be particular).
To my knowledge the reigning monarch can choose his or her Regnal name, much like the Pope. For example, George VI (of "The King's Speech" infamy) was actually named Albert and chose George VI upon his ascention to the throne after his brother's abdication. So I suspect it will be up to Charles, William, and the newest heir to choose their Regnal names at the time of ascention.
George VI took a different name to 1) provide continuity in an unstable time (his father was George V; his brother (Edward VIII) abdicated after a year as King to marry a twice-divorced American, just as we were heading into WWI) and 2) because Queen Victoria's son Albert Edward memorialized his father by suggesting he should be the only titled Albert in history, so he took Edward VII when he ascended and so this Albert also obliged (George was his fourth given name as it was).
Actually Victoria was Alexandrina Victoria, but was always called Princess Victoria, so when she ascended she kept continuity and chose Queen Victoria.
(I've been doing some reading on this over the past few days...)
I think they usually use one of their given names, though; never heard of an English king reigning in a name he didn't have.
Hard to imagine Charles would become King George after having been known by the public for (what will be) 70+ years as Prince Charles. George III, however, is his favorite historical king-ancestor.
Just some old encyclopedia entries at the library, GrayCat. I read up a bit on it after I saw "The King's Speech" (yes, I'm a bit behind the times) because the abdication of Edward VIII and the ascention of George VI was very interesting to me.
vc10um, I found a blog called The Anglophile at blog.londonconnection.com (i have no idea how to do links). He has great old pictures and
a lot of history, and some very interesting posts on Queen Mary and her jewels. I, too, am fascinated with Edward VIII and George VI!