How to use the phrase "the up-and-comers" properly? Is there any classification of age of people belong to "the up-and-comers"?
Actually, no. This does not relate to a particular age range.
It means someone new to a particular field who is making a name for themselves and is likely to continue to improve. Generally, this refers to younger people but the average age of the person actually doing the work is part of the concept:
Here are some examples and explanations
She's an up-and-coming child star.
In this case, she's clearly very young, likely between 7 and 16 years old. Because a child star must be a child, this person is obviously going to be on the younger side but "older" teenaged child stars are still regularly "discovered".
She's an up-and-coming young lawyer.
In this case, she's now much older than in the previous example. Lawyers are generally about 23-26 when they graduate from law school and pass the bar (in the US) so it's likely about someone in their twenties, though someone who finished school late (in their 30s or even later) can still be up-and-coming because they're still new to the field.
Similarly, in relation to doctors/specialists, they finish school around the same age.
She is an up-and-coming corporate executive.
Reaching this level of business often takes a good amount of time after one completes business school, so she's now likely in her 30s-40s.
As to up-and-comer, the noun form of the phrase is used but it can be a bit more clunky than using the adjective form, up-and-coming:
She's an up-and-comer in the world of corporate finance.
I'm sure there are other ways of using it but this is the first to come to mind.