I have been using Windows 7 Ultimate since it was in Beta, and used Inner Space on it daily.
To install IS on Windows 7 you may have had to -- at most -- right click it and select Run as Administrator. It will need to continue running as Administrator. If UAC is enabled in Windows, this means it should pop up a box asking you to confirm that it is allowed. Running in XP SP3 compatibility mode is not recommended.
To solve the DirectX problem, install the updates as instructed. The download you are pointed to is "dxwebsetup" which will download and install the latest DirectX updates.
DirectX 9, 10, and 11 are different products, and different games are developed to run on a specific version. Most of today's top games are built for DirectX 9. Each of DirectX 9, 10 and 11 comes with Windows 7 (Vista, likewise, has 9 and 10). DirectX 9.0c came out in 2004, but there have been a few dozen updates to DirectX 9.0c since then (1 every 2-3 months or so). Neither Windows Vista nor Windows 7 come with these updates.
There's a couple dozen DirectX 10 games, listed here:
http://en.wikipedia.org/wiki/List_of_ga ... 10_support
And then there's a few that can use DirectX 11:
http://en.wikipedia.org/wiki/List_of_ga ... 11_support
Many (most? all?) of the games that can use 10 and 11 also provide Direct3D9 mode. It would be silly for them not to, since they don't want to rule out Windows XP customers. When playing games with IS, Direct3D9 mode is currently required (though 10 and 11 support may be available in 2010).
That's not to say you wont reap benefits from Windows 7. The games should perform much better than on XP.