Suppose someone on another website or service strikes a similarity to a user of the actual Lemmy World instance and identifies as under 18 years old, are mods expected to investigate and enforce the policy of reporting and/or banning the user from Lemmy World based on a judgment call?

Mods aren't detectives. I doubt anyone would expect the mods of a community to dig THAT deep to investigate another user's posting history, especially not for ALL users in the bigger communities. I mean, you currently have 5.6k subscribers. Even if you'd spend only a single minute checking each account once a day on the off chance that some posts are similar elsewhere, that would amount to 94 hours a day, split between the 4 mods = 23.5 hours each day just checking comments. For each of you.

I'd say, unless the "other account" directly and openly states that Lemmy account XYZ is definitely theirs, and someone else directly reports it to you, I wouldn't worry about what they might or might not do on other instances. You're only responsible for your own community, not for the entirety of the Fediverse.

If their post is cross-posted to a community we moderate on Lemmy World, are we to delete the cross-post? Who would be in violation of the ToS because of the cross-post: the original poster or the cross-poster?

Technically, the crossposter would be the "guilty" part as the original poster has no say in whether or not other people crosspost their stuff elsewhere. They don't even get notified when this happens, so how would the OP be able to prevent it anyway?

However, the crossposter in question might not be aware about the Lemmy.world code of conduct if they're subscribed to another instance, or might not know that the post they just crossposted belonged to an u18 user, so you can't expect malicious intent. I think the best way to handle such a situation would be to notify the crossposter about the situation, explain that the crosspost is technically not allowed, and wait 24 hours[^1] or so to give them a chance to remove the post themselves. If that doesn't happen, delete the crosspost.

The only time I would issue an actual warning or maybe temp ban, is when it happens over and over again with the same user(s). Punishing someone for making an honest mistake is wrong, but if the person was informed about the post not being allowed and they keep doing it anyway out of spite or because they don't care, then that's a different story.

[...] if a user that is registered on another instance is under 18 years old and posts in a community or comments on a post that is originally from Lemmy World, that user is technically not accessing the site directly

4.1: No one under 18 years of age is allowed to use or access the website.

They might not have "accessed" the actual instance, but they ARE interacting with the content hosted on it, thus "using" it. But just like in the example above, there might not be any malicious intent involved. They likely just didn't know better, so I'd say tell them about the situation and politely ask for them to remove the comment(s) and not interact with the community again until they come of age. Immeditaly deleting posts and comments and/or hitting people with the ban hammer when they might not even have realized that they did something wrong in the first place ... that only breeds resentment.

The information from Lemmy World was forwarded to them, and the information they shared was forwarded to Lemmy World

Just a drastic example to put it into perspective: If someone isn't old enough to buy alcohol, and another person buys alcohol FOR them, that doesn't suddenly make it okay for the underage party to consume it. And if Lemmy.world content isn't meant to be "consumed" by an underage end user, it should not matter HOW they got the content in the first place. The shop clerk however did nothing wrong as they didn't sell alcohol to an underage person, so it would not be "your fault" as a mod if your content was taken elsewhere by someone else without your knowledge. You're not omniscient =P

...but again: polite warning and explanation first, always. Explain instead of punish, because they might not be aware that they even did something wrong. Give them the chance to fix their mistakes first, and only take action if they're unwilling to do that.

[^1]: It can even be longer. Not everyone is online every single day, so I usually just occasionally check whether or not they have been active on the site. No posts or comments since the removal was requested = they're likely not online and thus unable to react. Especially with such a comparably tame issue like the age restriction, it is not the end of the world if the post/comment stays online for a while ;) The content isn't harmful, after all.

[Edit: fixed some typos and added a footnote]

You don't have to go looking for evidence, but I think you're within your rights to ban any underage interactors you know of.