Magento Connect vs Magento Marketplace
Currently the Marketplace is Magento 2-only, and Connect is Magento 1-only. Later this year Magento 1 support will be added to the Marketplace (on a temporary basis, I think), at which point vendors will be engaged to migrate their listings across. Once that is done, Connect will go away entirely. The exact timeline and migration process is yet to be announced.
Yes, there's no harm to keeping extensions on both (although, as noted, currently there should not be any overlap in listings--one is M1, the other is M2).
As noted in #1, currently the Marketplace is M2-only, so this is not relevant. I would expect once Magento 1 extensions are supported on the new Marketplace, that will include support for the existing Magento 1 Connect Manager.
Source: Announcements and session on the Marketplace at Magento Imagine 2016.