I am one of the few non-profit organizations which was issued the old (no longer available) 5000 slot limit TeamSpeak 2 NPO license. Can I use this license with TeamSpeak 3?

No. TeamSpeak 3 will introduce an entirely new licensing system with new limits to non-profit entities. However, we do intend to allow NPOs to host what we consider to be a generous number of virtual servers and slots for non-profit reasons. For the few non-profit entities which still have a TeamSpeak 2, 5000 slot limit NPO license, this type of license will simply not exist at all in TeamSpeak 3...nor can your TeamSpeak 2 NPO license be "transferred" to TeamSpeak 3.