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No, however you can purchase a Mac version of Illustrator if you want.
The operating systems that are compatible with Adobe software include Windows 7, Windows 8, Mac OS X, Mac OS X Snow Leopard and Mac OS X Mountain Lion.
Mac "Lion" and Mac "Snow Leopard" refer to iterations of Apple's Mac OS X (operating system, version 10). "Lion" is the newest as of this writing, and is version 10.7, while "Snow Leopard" is its immediate predecessor, and is version 10.6.
Snow Leopard is an upgrade to Leopard and so you will need Leopard installed on your Mac before installing Snow Leopard. For Mac computers with an Intel processor running Tiger you will need to purchase the box set that includes iLife 09 and iWork. (See links below)
There is a version of Illustrator for Windows, so yes you can use Illustrator on a Windows machine. You can't use the Mac version of it on your PC, so if you bought a copy of Illustrator from eBay and it was the Mac version, you'll need a Mac to run that.
10.5, Snow Leopard is 10.6
Flash is an adobe program just like Photoshop and Illustrator. It is the same on both mac's and pc's.
Mac OS X 10.6 (Snow Leopard) and all earlier versions of Mac OS X will support networks.
All new Macs are supplied with the latest version of Mac OS X which is currently 10.6 Snow Leopard.
yes
Apple's Mac OS X 10.6 (Snow Leopard) can be installed on an Intel laptop if the laptop is made by Apple.
Although the appearance of Mac OS X 10.5 (Leopard) and Mac OS X 10.6 (Snow Leopard) is similar there are many differences in the workings of the two systems. Most notably Snow Leopard requires a Mac with an Intel processor whereas Leopard will run on earlier Macs with G4 or G5 Power PC processor as well as the newer Intel based Macs.