Is there any way to set the display name of the application which is shown on iPhone and on iTunes different? The name that I wrote to "Display bundle name" in info.plist is shown both on iTunes and iPhone, is there any way to separate them?
My second question is: how can I change the genre of my application? Now it's "Unknown Genre".
The name that is displayed on the iPhone beneath the icon is set in the "Product Name" build setting for your apps target. You can set this to whatever you feel appropriate.
Then whatever name you entered as the App Name in iTunes Connect is shown in iTunes and the App Store.
Apps don't normally have a genre. Apps are separated into Categories, of which you should have been able to pick 2 (primary and secondary) during your submission through iTunes Connect.
You should be able to update these category choices by logging into iTunes Connect and editing your app.