This is rather encouraging to hear, particularly the bolded elements. I think I'll give it a chance based upon this description. An introspective, meditative story set in what is a largely grounded world seems much more appealing than what I've previously seen, and the anime being a faithful adaptation is particularly encouraging.
Regarding what I meant by 'age inappropriate characters', I was referring more so to a lack of congruity between a character's age and a character's actions rather than a gratuitous sexualisation of young, typically female characters (although that admittedly is something I'm not a fan of either). Based upon what you've said, I'm quite sure this isn't really an issue or prevalent in "Monster", but to provide further clarification with an example, I'd cite the rebellion (to be broad) in Code Geass, where almost every influential member of the rebellion, and two in the established state, is of school-going age (or in Death Note where the two smartest people alive are both just entering college, one of those having already established themselves as the world's greatest detective). While this is often flimsily justified in the plot, and/or is a crucial component of the overall story, the characters' actions, motivations, and position does not seem to allign with how old they are, and removes some of the authenticity to me.
I should probably just clarify that I'm not trying to say that characters who break stereotypes are problematic to me, only when it is done in such an extreme manner with little confrontation or justification. I'm unsure if I've adequately clarified what, specifically, I'm trying to comment upon here, so I've searched TV Tropes and located this
here which refers to what I'm trying to express my distaste for.