Luke,
1) no problem. The code is GPL or if you want the Elgg code also to be own labeled you can use the MIT licensed variant
2) Bad idea. Iframing is always an option, but passing credentials via the iframe url is very unsecure. Integrating with the other site should be handle by the clientsoftware or by a process like OpenID or oAuth
3) The subnetworks can be the group functionality of Elgg. Elgg access rules will prevent you from seeing content not shared with you and you can limit membership to groups by groupadmin approval. This there is no accounting problem or shared feature issue.
So, yeah this can be done with Elgg :)
ad 1) Elgg 1.7 is under GPL and MIT, Elgg 1.8 is under GPL only right now, although there's a thread here to modify it to MIT-compatible
As Jeroen pointed out, Elgg can be used to your project fairly easily.
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