Do Actors Have To Match The Ethnic/Race Of The Characters They Play?

Some people don’t like the idea of an American playing James Bond, they want to keep Bond played by any actor from United Kingdom area (England, Scotland, Ireland, Wales)

I still haven’t heard anybody complaining about Charlton Heston playing Moses in Ten Commandments when Heston was not even a Jew.

People didn’t mind Robert Downey Jr. playing an Australian who’s also playing an African American, hardly anybody thoughts it was stereotypical, he even received an Oscar nomination. I thought his performance was brilliant by the way.
In fact Tropic Thunder was criticized more for its humor on the develop-mentally disabled.

When Ziyi Zhang got to play the lead in Memoirs Of A Geisha, I heard some in the Asian community consider it not cool that Zhang, a Chinese, was playing a Japanese character.

There’s a debate out there whether CAPTAIN AMERICA in the upcoming Marvels’ live action movie, can be played by a Non-American actor

I know for sure that some fans think Jake Gyllenhaal doesn’t look Persian therefore PRINCE OF PERSIA the movie is going to stink because of that reason alone.

Christian Bale and Russell Crowe playing American cowboys (3:10 To Yuma), Paul Bettany playing a Southerner (Secret Life Of Bees), Michael Caine for his New England accent (The Cider House Roles), these actor received praises for their performances.

I’m Asian and a long time ago when I heard that Clint Eastwood was looking for people to play Hmong in Gran Torino, I contacted the casting director and she said they’re only looking for Hmong actors, which I find odd because I could swear that some of the actors playing Japanese soldiers in Clint Eastwood‘s Letters From Iwo Jima were Koreans and Chinese, not all of them were Japanese.

So how do you decide which ones you’re okay with and which ones you think is offensive?
Do you let it slip by if the character was played by a great actor like Anthony Hopkins (Nixon, Amistad)?
So the actors have to really really really try look the part to deserve audience’s approval?

Maybe PRINCE OF PERSIA crew didn’t do a good job of prepping Jake to look more Persian-like? If Jake did some pitch perfect Persian accent, then would that make it acceptable?
Does it make a difference if they’re playing real life or fictional characters?
Do Actors have to match the ethnic/race of the characters of they play?