How can I do an advanced serach to find those opportunities that have been
neglected e.g. where an opportunity does not have any activities open that
are due in the future OR Where an opportunity does not have an activity
modified in the last X.

If this is not possible can anyone help me out, what do you do???
Potentiall I was thinking about adding a field to opportunity that gets
updated whenever an activity associated with the opportity get created or
modified. Is this possible without using callouts (I am not a developer).

Thanks James