Wowie, thanks for the ask my liege! I appreciate this opportunity.
Really really. ^^
My self inserts all share the same name, Eli.
Also oh god, hope you're ready for an essay and a half.
🌎 - I'll answer this one for my main 3 inserts (if I listed them all we'd be here all day).
The first doesn't really have a 'world' per say, as the F/O in question (Evelin) is an OC of a friend. Probably as close to 'non-fiction' as I'll get. For this ask we'll refer to him as Normal Eli.
The second however is from Honkai Star Rail, and he's the 'Co-Conductor' of the Astral Express, being an assistant to Pom-Pom, and a sort of apprentice to Himeko. He's simply called Conductor Eli.
The third is from Fire Emblem (Birthright and Conquest specifically). He's a soldier of Nohr, and doesn't talk much, if at all. I just call him Nohrian Eli.
There's alot more than these (Fate Grand Order, The World Ends With You, Bungo Stray Dogs, Hypnosis Mic, Persona 5, and much more) but again, we'd be here all day
Given that the last two dabble in alternate timelines and dimensions, it wouldn't be hard to imagine a scenario where they're pulled into a sort of Injustice-style plot, being brought together to halt some event or threat.
As for the first...probably isekai, as cliche as it sounds.
🫂 - This is a tough one. Eli typically doesn't have a high opinion of himself, to the point that there's a running gag and plot point in the fics my friend writes that use him.
That being that he tends to not like people who share a similar mindset or personality traits. That being said, some Elis would be the exception.
Biggest example being Conductor, whose much more open and friendly than Eli typically is. He still has the usual quirks (a knack for sarcasm, tenancy to be sassy or cheeky, and low confidence), but he's much less bogged down. He'd definitely try and get along with the other 'hims', and probably try and mediate. Some Elis might find him annoying though, due to him being a bit of a fluffball when compared to the rest of them.
A much different vibe when compared to someone like Nohrian, whose the complete opposite, being closed off and is resistant to forming bonds with people. He wouldn't be hostile though...more like the guy who shows up for class projects, but you never see him outside of that. Also again, he barely speaks if ever.
Most other Elis follow Normal's modus operandi. That said, most would be reluctant but would eventually get used to each other.
🌐 - I can definitely see most of them keeping in touch after going home if they could. A multiversal chatroom most likely. Maybe even visiting each other. I know Conductor would, his curiosity and eagerness to learn being piqued when hearing about how the others' home worlds are.
Nohrian, coming from a fantasy setting, would need to rely on visits because...he doesn't have a phone, nor does he know what that is. But even if he did have a phone he'd probably have everyone on 'read'. Both irl and online, he doesn't speak.
The others would communicate decently regularly, and maybe even help other with advice if a situation calls for it. Visits would depend on their schedule because it varies wildly between versions.