Open new Terminal in same SSH session

Yes, you can reuse exiting ssh connection and open ssh in whatever terminal you like. See this answer to a StackOverflow question for details:

If you open the first connection with -M:

ssh -M $REMOTEHOST

subsequent connections to $REMOTEHOST will "piggyback" on the connection established by the master ssh. Most noticeably, further authentication is not required. See man ssh_config under "ControlMaster" for more details. Use -S to specify the path to the shared socket; I'm not sure what the default is, because I configure connection sharing using the configuration file instead.

In my .ssh/config file, I have the following lines:

host *
  ControlMaster auto
  ControlPath ~/.ssh/ssh_mux_%h_%p_%r

This way, I don't have to remember to use -M or -S; ssh figures out if a sharable connection already exists for the host/port/username combination and uses that if possible.