The age thing is between the user and Fandom. You are under no obligation to take any action. You're free to do so, but you won't be held accountable if you look the other way.

You also don't have to be nicer to them, because you think they are below the allowed age. Sometimes short blocks can have more influence than any explanation.

Luckily most learn and grow. At least enough to contribute to the wiki at a descent quality. I'm currently working on template pages which should make it more easy to add new pages by copying the templates. On multiple pages of the wiki is stated that if you need help, you can contact the admins any time. 

I'm still thinking about what to do about the users who do want to contribute with good intent, but don't feel like learning.

That's a good thing to know. Thank you as always!

I'm really strugling with this at the moment and in my eyes I often feel I'm less nice to the younger users than the older ones. Or at least less patient.

This is mostly due various different incidents with young users (vandalism, spam, (pre-)teen drama's. Most of them got banned btw). Sometimes I really have to remind myself I'm talking to young people and that they can think totally different, which is not wrong.

But over the past five years I and a bunch of other users put a lot of time at our wiki. It's a fandom full of stories and a fantastical place with wonderful young writers (I love to see their stories grow), but then you have these (very) young users who can't even be bothered by starting their sentence or writing their name with a capital and end it with a period. It hurts.

I know this young writer in person and she seems to enjoy writing from time to time, but she really doesn't care about anything else (grammar, punctuation, etc).

So yeah...I'm a bit bummed.

Well, a good compromise is to let them make those mistakes in a user blog or a user subpage. There can be different rules for different parts of the wiki. In shared pages, it is good to have standards... you shouldn't feel bad about that. There are many admins about a thousand times as strict (often irrationally so).

As has already been mentioned, you are not required to report underage users. However, I would like to make the following additional points.

1. The age is 16 for EEA and California, USA. For users from any other part of the world, the age is 13. So a user that is 14 is not necessarily underage.
2. Even if you "remove" the private information, it could still accessible via the logs. Staff have special permissions for dealing with this stuff. Unfortunately, that would require you to tell staff about the user which would probably result is them getting globally banned.

Plus if u use email its 13+

The underage issue is a concern, but not the main problem.