Not exactly, Johnny, but you are close. The kernel is the core of the operating system. As such, it performs the functions you describe (controlling radios, network, user input input, camera, etc...) by talking to device drivers, which in turn control the hardware. It is also the base layer that the Android or IOS operating system runs on. If you think of the CPU as the heart of a physical computer, the kernel is the heart of the software. It isn't part of the ROM, it resides in the ROM. The ROM (read only memory) is a place where important system programs (device drivers, IOS or Android, and the kernel) reside when the device is powered off. In this modern age, kernels can be updated. This is known colloquially as "installing a new ROM", although you aren't changing out any physical parts. What happens when you "install a new ROM" is that you place the device in a state where the ROM can be overwritten with a new kernel, a new operating system, or new device drivers.
People who are very good programmers (such as the RedSn0w team, or the Apple IOS team) are capable of modifying the kernel and device drivers to add or remove functionality from the device. Since Apple doesn't want anyone but them to modify the kernel, the device must be "jailbroken" to allow 3rd party kernels to be installed. Google has no such hangups, although the carriers that get the Android OS and kernel from Google do, which is why Android phones must also be jailbroken if you want to install new functionality.