Does Apple have a way to find out that an application already is in the Cydia store when I try to submit it to the AppStore?
For Example, if I change the name and icon for an application that appears already in Cydia and try to submit it to AppStore? Can Apple find out?
Is it possible that Apple collects statistics about jailbroken phones and their applications and has this data in its db (by sending it in some background process)?
First of all, I'm assuming that the existing app on Cydia is your app. I certainly don't condone stealing some other developer's intellectual property, just because they released a Cydia store app.
To answer your questions:
Yes, of course. They can just use Cydia themselves and search for your app's name. They could certainly also just use a desktop browser and search for your app's name, to see if there is any obvious copyright infringement. While I don't think Apple feels obligated to do this, I have had apps rejected (before ever being released) because Apple believed my client was violating some other 3rd-party's intellectual property. This kind of search could also turn up a listing in a Cydia store repository. So, it is possible.
I very much doubt Apple would find out about this. Simply renaming the app seems likely to avoid any problems.
I'm not sure I understand the question. Are you asking if iOS checks to see if the device it's running on has been jailbroken, and if so, reports that status back to Apple? I don't know, but millions of users run jailbroken phones, and I haven't seen Apple try to disable their functionality.
I have had clients release apps on the Cydia store first, and then try to release them on the iTunes App store. Although none of those have been approved by Apple, they have been rejected for private API usage, or violation of Apple's Human Interface Guidelines, not because they already existed on Cydia.
I suppose that if an app was released on Cydia, because it used private APIs, and then iOS later added that capability in public APIs, then you could try to submit it to Apple.
I don't work for Apple, so these are only my guesses based on personal experience.