Colloquial America = USA. For example: I want to study in America = I want to study in the USA. When it comes to the continent, they say Latin America, South America or North America. For example: Genetic analysis determined that the Indians of North and South America are descendants of the Buryats from Siberia.
Russians, much more often than not, write ё as е in everyday life, signs, newspapers, etc, that is, they completely omit the "little dots" and write it just like normal е. Both are however different sounds and letters in modern Russian, and they tell them apart by context and by knowing the word (For example, when seeing Петр they would automatically read it as Пётр).
ё is written with dots only in foreigners language books (like in Duolingo :P), in children books and material, in some books, and when there is a real ambiguity without writing the dots (I am not sure I have seen this, but I hope they do write всё instead of все when meaning всё... Though I think they still just write все most of the time and use context to distinguish them).
My understanding is that this is historical: Originally the e sound was used always, and ё as a letter and sound did not exist at all. Speakers started pronouncing stressed е as ё in some words following certain patterns, but since ё did not exist, it was still written as е. Eventually ё was introduced as a written letter, but by then everybody was already accustomed to writing just е, and a lot of books and already written material was like that.
Writing ё always as ё (which feels more natural to foreign learners like us) is correct Russian (and obviously, omitting it as Russians always do is also correct).