Does an enumerator print the first occurrence of a word in finite time? For a proof I need to use the fact that every word in the language of an enumerator occur on the output paper in finite time. Is it true?
For example, the language of the natural numbers in decimal representation. Can the enumerator print the odd numbers first and then the even numbers? (if yes, I am wrong)

$1, 3, 5, 7, ..., 2, 4, 6 ,8, ...$

As I know, we get to the even numbers but not in finite time (maybe transfinite induction based on this)
Your interpretation is correct, and the easiest way to look at it is to know both definitions of recursively enumerable sets:


There is an algorithm (potentially running forever) that enumerates the members of $S$.
There is an algorithm for which the algorithm halts only on elements of $S$.


These two definitions are equivalent, and if you use the second one then you can easily answer your own question.

However, do note an important caveat. Although for any given word $w \in S$ there is some time $T_w$ such that after $T_w$ steps, $w$ will have been enumerated. However, there is no way to bound $T_w$ from above as a function of $w$. It can grow faster than any computable function.