This is definitely doable. Sublime Text already does something similar with the Find Results page.
Remember, the console is just a view that you can easily show/hide. It's the default debug output of pretty much everything and a bit overloaded. You might consider using a view for this instead of a console:
output = sublime.active_window().new_file()
output.set_name('Server Output View')
By setting an arbitrary setting like 'server_output' to True on your output view, you can check if the current view has this setting when determining whether to act on an event.
You can easily insert text into this output view using my Edit class:
from .edit import Edit
with Edit(output) as edit:
edit.append('new line of stuff\n')
I implemented a double-click action in SublimeXiki:
It manages to not break double-click-drag using a bit of hackery.