You can SSH to your server by clicking File -> New, then click Terminal (under tools).
This will prompt you for your server hosstname and username.
Key authentication is supported. You can generate an SSH key from your account area.
Put the public key in the ~/.ssh/authorized_keys file on your server.
You will then be able to log in without a password.
To copy a mouse selection:
Ctrl+A - Ctrl+C or Ctrl+Insert
Ctrl+A - Ctrl+V or Shift+Insert
To copy and paste a mouse selection:
Ctrl+A - Ctrl+C - Ctrl+V
Ctrl-A enters prefix mode, from here you can type Ctrl-V to paste. Press [ in prefix mode to enter selection mode. To select text press v (or space) to enter visual mode, use hjkl to navigate and create a selection, and press Ctrl-C to copy.
Ctrl-C (in visual mode) and Ctrl-V (in prefix mode) should work in any OS for copy and paste. y (in visual mode) will work for copying only on X11 systems. It will copy to the primary selection.
Note: Ctrl-C will also work in prefix mode for the regular OS/browser selection. If you want to select text with your mouse and copy it to the clipboard, simply select the text and type Ctrl-A + Ctrl-C, and Ctrl-A + Ctrl-V to paste it.