Translation:I travel to America twice a month.
Twice a month to America from the Netherlands, regardless of whether you assume America is the US, sounds brutal to me.
In a sentence like this, does Amerika ever not refer to the US?
Yeah it could just refer to the continent, though probably a lot of people would most likely think that you are referring to the US.
Which continent? North or South America?
From El2theK's comment I assume Dutch, as many other Europeans, just consider North and South America to be subdivisions of one single continent.
I think it's discriminating the rest of the countries that are a part of America as a continent. I believe it'd be a lot better if in all of these sentences America was changed to Verenigde Staten.
I don't agree with you. The suggested translation says "America" as well and not "the USA".