Parent/Child Relationships with Multisites?

Yes – this would depend mostly on the theme. You would likely need to create a custom theme or a child theme which would present menus in the way you’re requesting, and you would want to go with a subfolder install vs. a subdomain install.

I assume www.mainsite.com/multisite would be the “main” / parent MultiSite, so all subfolders of /multisite/ would be separate subsites. If this is not the case, you might have to go to a MultiNetwork setup or perhaps consider a single install with more of a focus on permissions by CPT rather than separate subsites.