It is possible that your server is configured with a different keyboard layout than your chromebook (Droplets default to an EN-US standard keyboard). This will not matter when using SSH but when using the web console the key events are sent directly to your server (and your server see's it as a local display and keyboard). You can re-configure the keyboard by running:
on your Droplet.
You also mentioned that you are able to use the console but not SSH from your location. SSH by default runs on port 22 so it would appear that is blocked. The control panel runs on standard web ports but the web console utilizes a random port (usually between 6000-7000) to establish it's connection. Since you can use the console that would indicate that your network is only blocking specific ports. If you change the port that your SSH service is using you may be able to use SSH instead of the console.
Edit the file
and change "Port" to any other port.
Then restart the service with:
service ssh restart
and update your ssh client's connection configuration to reflect the new port.