You definitely need the first two. The last is optional: ... intelligent, American friends
suggests you have some other single and handsome and intelligent friends who are not American. ... intelligent American friends
suggests you have other American friends who are not single and handsome and intelligent.
Partly taking up ewie's explanation...
We might well say, "My friends are single, handsome and intelligent". When we put this series of adjectives before the noun, the "and" disappears and is replaced by a comma.
We are less likely to say, "My friends are single, handsome, intelligent and American". "American" is not part of the list of good qualities. We are more likely to say, "My American friends are single, handsome and intelligent". Placed before the noun, the list stops with "intelligent" and there is no need for a comma before "American friends".
It's like "The beautiful Spanish girl had long, dark hair".
We say "Her hair was long and dark", so these are equivalent adjectives in a list, requiring a comma when coming before "hair".
What about "The beautiful Spanish girl"? We are likely to say "The Spanish girl was beautiful" or "The beautiful girl was Spanish". We are less likely to say "The girl was beautiful and Spanish". These two adjectives do not form part of a list. That's why I put no comma.
This is my general rule: put the adjectives after the noun and see what you'd normally say.
I once contested the sentence, "Brooklyn Bridge is a 19th century, engineering marvel". We wouldn't say, "the marvel is 19th century and engineering".