Why did they pick an American actor to play Sherlock Holmes?

Because they liked both his star appeal and his acting for the role?

Superstar actors get to play whatever roles they want. They cast Kevin Costner in the very British role of Robin Hood, for goodness sakes. At least Downey did a decent job of maintaining the same accent on consecutive sentences.

This is a Hollywood film, and Hollywood is rife with, shall we say, culturally insensitive casting decisions to get stars in roles? That's well known as "whitewashing," so it get much worse than this. Mickey Rooney's Mr. Yunioshi from Breakfast at Tiffany's is the most famous and appalling example of this, I'd say. Christian Bale as Moses has been criticized, where Charlton Heston in pretty much identical casting wasn't. Times have changed.

However, here are examples of, not so much whitewashing, but roles where the actors are chosen to play roles not of their ethnicity.

• Brad Pitt as an Irish guy Devil's Own.
• Tom Cruise as an Irish guy in Far and Away.
• Any movie set in ancient Greece or Rome - British folk
• Any movie set in medieval times - British folk, or sometimes people with French accents
• Renee Zellweger as Bridget Jones..... three times!
• Many of the cast of Walking Dead, including Andrew Lincoln, are British.
• Alan Rickman (British) as German terrorist Hans Gruber in Die Hard
• Jeremy Irons (British) as Hans Gruber's brother in Die Hard 3

However, it's called acting, so, if the actor is up for it, it's not that big a deal, as long as it is plausible. The casting of Jonathan Pryce was roundly criticized as an earlier version of whitewashing when he was cast in the Broadway production of Miss Saigon. He's such a great actor, though, that he won many awards for that role. So, would it have been better if they cast an Asian in that part? From a purely entertainment perspective? Can you separate that, today, from cultural sensibilities? That makes for interesting discussion.

Aside from the good answers from a casting point of view, you're simplifying a lot here when it comes to American vs British. The Sherlock Holmes stories took place from about 1880 to 1900. Cultural habits, accents and so on will have drastically changed even within Britain itself since then. For authenticity there's no reason to assume a British actor would do better. Sure, he may sound more British, but that's modern-day British, not Sherlock Holmes' British.

You could even go further: American Independence was 1776. This means that the stories are about as close to the point where American and British cultures started diverging as they are to modern day. Depending on how language evolved it could even be the typical modern American Accent comes closer to the 1880 British Accent than a modern British accent. I know it's unlikely and you'd need someone specialized in English language history to confirm/deny that. I just want to point out the flaw in the 'purely British' reasoning.

So, it doesn't matter, as Sherlock Holmes is a 19th century British character and there are no 19th century British people alive today to portray one.

Why did they pick Daniel Day-Lewis to play Abraham Lincoln in Steven Spielberg's film or Anthony Hopkins to play Nixon and the American war hero in Legends of the Fall?

There are more cases where British actors played American roles and vice versa. Casting seems to be more focused on an actor's popularity and acting skills rather than their nationalities and accents. In addition, even though Sherlock Holmes is a purely British character, the story is internationally well-known.

One more important factor is the US market is far bigger than the UK market and as a producer, choosing an American actor for the role doesn't seem to be unreasonable.