$\mathbb {R,Z}$ etc. are imitating the way we write bold R, Z on a blackboard (hence the name, blackboard bold). It can be argued that when TeXing (not actually writing on a blackboard), you should write $\mathbf {R,Z}$ instead (since that's what $\mathbb {R,Z}$ are meant to represent on a blackboard, in the first place!), and I, for one, do just that most of the time.
They are written in bold to make the name distinct, because $R,Z$ may be used to represent other, more locally defined objects, while bold letters are rarely used as local variables. As to why are the particular letters are used, the $\bf R$ is probably self-explanatory, while $\bf Z$ originates from German (Zahlen).
$\bf K$ as a dummy field name also comes from German (Körper), and in this case bold is likely used to imitate $\bf R,C$ and to indicate that it is "the" background field when it is fixed in the context, so it is, at least locally, as fundamental as $\bf R,C$ are (e.g. in linear algebra and algebraic geometry). It is less often used in that way when we consider many distinct fields ard rings, like in abstract algebra (where letters starting with $K$, and continuing with $L$, and sometimes $M,N$, are still often used to denote fields, but are rarely bolded).
The set $\mathbb{N}$ is the natural numbers. But usually in the context of number theory when one writes $n$, one means a particular natural number. For example, $n$ could be $5$ or $12$ or $1$ or $1,234,453,564,234,134,179,200$. But they would usually not use it to denote any integer and certainly not a real number.
In any case, what the variable $n$ is being used for should always be stated somewhere in the text whenever it is used, so search for what they have declared it to represent.
Best Answer
To summarize what has been said in the comments, there are no "official" symbols. Use whichever notation you feel most comfortable with, as long as it makes sense and can be easily understood by the general audience.
Some examples include:
$\mathbb{Z}_{\ge 0},\mathbb{Z}^{+}\cup\{0\},\mathbb{N}\cup\{0\},\mathbb{N}_0$
Also note that because of different conventions, what you refer to as "whole numbers" may or may not include zero. From Wikipedia: