Except the companies don't see any of the money once the IPO is finished.

Of course, the only reason the IPO can proceed is because people who buy shares know that they can sell them to other people later. So I suppose in that sense they help. However, once the market already exists, no extra value is created by an extra trader entering it.

As others have pointed out though, there genuinely is value created by the options trading market, since the existence of options allows risk to be spread.

Well, companies do sometimes make further public offerings.

Stock trading also provides a somewhat objective measure of the value of the company. For instance, if MS wants to buy Yahoo, how much should they offer? Without the stock market, it's tough to tell if they are lowballing or paying a premium.

You're spot on that the market is a means of both creating information and disseminating information. Information is key to game theory. That's where sigma algebras and filtrations start to creep into theory.