I see that your Mac OS is 10.5.8; does that code type 'x86' mean it's an Intel?
If so, you need to download version 10.3.183.90 from http://helpx.adobe.com/flash-player/kb/archived-flash-player-versions.html#Flash%20Player% 20archives
You can download MacOS 10.9 from Apple for free: Apple - OS X Mavericks - Do even more with new apps and features.
The oldest version of MacOS we support is 10.6, and the vast majority of our effort and testing is happening on 10.9 and 10.10. I'd highly recommend taking advantage of the free upgrade, as Apple is pretty aggressive about dropping support for older operating systems in their development tools and SDKs. This makes it very difficult to support older Macs indefinitely.
As Flash Player 10.3 is literally missing hundreds of security fixes, I would not recommend that anyone use it. We provide it for developers testing their applications for backward compatibility, but our strong recommendation is to run a current version and set your system to always accept updates from Adobe. If you're not automatically accepting updates, you could be leaving yourself exposed to security threats that we've addressed in the latest versions.