Manners in the sense of mores isn't very widely used now. Winchester College's motto 'Manners makyth Man' is much misunderstood - people think it's about good manners.
One of the few ways in which manners would be understood in this sense is the phrase you mention: American culture and manners. Then you say etiquette (it isn't usually used in the plural) and I wonder if you mean good manners, rather than mores (customs, ways of doing things), in which case it's odd to put those manners alongside culture.
My advice would be as follows:
1. If you mean manners (how to behave) say good manners.
2. If you mean manners (mores - customs) say customs.