Can I host my own gitlab instance and tell github users to pull from my gitlab instance? And can they do it with the usual interface that they use for closing github pull requests?

Ah, ya, you aren't going to be able to use Github's users on a Gitlab instance. It is like hosting your own copy of a Github-clone, basically. :/

You could write API integration between the two yourself but it wouldn't be that similar.

Imagine if we had to beg Google to fix Gmail if we wanted to email people our own way. :-(

Sorry my suggestion didn't help you, I just thought given your objections it might.

Ya, I host a GitLab instance for personal and one for work so I don't really have to deal with GitHub beyond when I am looking for open source projects.

Part of why my profile there is so empty :/