Is it possible for a BLE device to be connected to an iOS device using Bluetooth classic (via settings) and also be connected to the same device using BLE via an application?
Yes. You can for example use A2DP music streaming by connecting via the Settings menu and establish a concurrent BLE link from your app. Some versions of BT chips have restrictions, though, that prevent you from being a BT Classic Slave and a LE Peripheral at the same time. For these chips, BT Classic Slave + LE Central should work fine. From the iPhone side, it's no problem.