The two main groups jockeying for position are EOS BIOS Boot and EOS Core. The Jungle Testnet is being run by another team, the Crypto Lions block producer candidate: http://cryptolions.io/

There's no possibility of the Jungle Testnet becoming the mainnet, it is and shall remain just a testnet. But we like it as a way to play around with EOS without worrying that your actions will affect the mainnet. It's perfect for development work and practice.

Yes, BuildTeam is setting up another test node on the EOS BIOS Boot network. We aim to be well positioned for the mainnet launch regardless of whichever chain is eventually chosen to become the mainnet.

You can use cleos to connect to a remote node on any network that has made itself publicly available. However, you also still need keosd running for wallet management. And bits of code in cleos and keosd have likely changed a good deal since Dawn 3.x. To be safe and not have to worry about any compatibility issues, it's best to just build the entire thing again.