Is Mac Mini enough to publish my game on iTunes or do I need iPod or iPhone to publish the game on iTunes?
As a user I would hope that you have run the app on the IOS device that Iuse. Thus iff you say it runs on an iPad you have run it on an iPad, if I have an IPhone you have run it on an iPhone etc.

If I have a problem I expect you to be able to reproduce it on the hardware I have or be able to say that you see no problem on the hardware.

No simulator is perfect there will be differences between it and real running - especially as they use different types of processor etc. One example of this as pointed out by 0x7fffffff is that the file system in the simulator is case insensitive but on a IOS device it is case sensitive

So I would expect professional support to have run the app on at least an iPad and an iPhone and preferably on various models e.g. iPad and iPad mini, iPhone 4s and 5.