"Could a system be set up to perform web related actions based on input from the web itself"
The idea being that I set up a series of actions. I then define a trigger, the trigger being something which the system managing this can monitor and watch in a fully automated manner, when conditions are "right" my action or actions are initiated without further intervention on my behalf.
Some real case studies to illustrate this idea.
I hold a portfolio of investments. For purposes of simplification let's suppose that my portfolio is relatively simple. However I have an action I wish to execute, in a fully automated manner, but which is significently more complex than a simple order I could leave with a broker (sell if the price hits $X)
So what I might want is something like "If the price of Oil goes over $75 sell each of transportation stocks I own IF that firm does not have more then y % of it's projected sales from hybrids or other near zero emission vehicles."
Or, if I have not posted an entry on my primary blog for more than Y days (or hours), check my publically available calendar for scheduled events, if it is a holiday OR if I have a scheduled event, post to my blog a note of the form "Shannon Clark is on holiday to the .... conference in Florida, will return on ... date"
Or, if my public calendar shows me out of town for more than a full week, contact the magazines I subscribe to (pulling that data from somewhere I publish it), checking first that they have an issue expected to be delivered to me while I would not be home, and arrange for that month's issue to be either held at the post office until my return, or held and my subscription extended. Likewise take care of other regularly scheduled events/deadlines I might otherwise miss.
Many of the ideas you sketch are what is done by publish and subscribe systems... See: http://pubsub.com/ for one example of a simple "trigger" system that is used to filter new information posted to the blogosphere or other networks. When you subscribe to searches on PubSub.com, you're basically asking that a trigger be created...