Not sure if I agree that deleting cookies for apple.com from Safari, could affect iTunes. iTunes most likely uses a security token stored in the user's Keychain, rather than using a cookie. Cookies would not serve any useful purpose for iTunes the way they are used for web browsers (caching credentials, login status, and tracking you for advertising/marketing purposes, etc) - iTunes already knows who you are due to the fact that it knows and uses your iTunes Account credentials every time it talks to Apple's servers. Using cookies to cache your login status would be leaving the door wide open to fraud, so there's no way they'd use cookies for that. Tracking for marketing/advertising? No, because they'd just use your iTunes Account for that purpose. I'm not saying I know the inner workings of iTunes, but as an IT professional I cannot see any best-practise reason why they would use cookies at all, let alone share the same cookie file as Safari.
Besides that, the two are completely separate, unrelated applications, and not only would it be bad software practise for two unrelated applications sharing cookies, it could be problematic for two apps to be reading & writing to the same file.
Regarding the fix to the blank web page issue - I have just tried the suggested fix to this issue - deleting apple.com cookies from within Safari - and I can confirm that it resolved this long-standing issue. For me, the issue had persisted across Mavericks, Yosemite, and then a complete wipe and reinstall of Yosemite.
If you really think I'm wrong about the cookie-sharing theory, then I would suggest that an easy way to prove your theory would be to create a new user account on your Mac, log in under this new account; then ensure the Safari cookies folder is completely empty, then set up iTunes with your iTunes account, then re-examine the cookies folder.