Can a non-rooted Android 4.0.3 smartphone’s ROM be backed up?
To answer your first question, you cannot do a whole image backup from within Android without root, as no application or service you can run will be able to read the system partitions.  

If the "recovery" console (a small, BIOS-like image installed on the device) contains backup support, it has access to the raw partitions and could create a "Nandroid" backup image (see About the android recovery mode).  If your phone's default recovery does not support image backup (it would be somewhat surprising if it did), you may be able to install a new recovery, without "rooting" the Android installation.  Booting into recovery is very device specific (e.g., hold down power and volume down), and its features (and replacability) are fairly specific too.

If you have a recent Android Development Kit (ADK) installed, you can use the adb backup (and adb restore) commands to backup your installed apps and (some of) your data.  But its not a complete image/ROM backup.  I can't find any official documentation, but this   XDA Dev Forum post goes over it in some detail.

To your second question, see: How do I root my Android device?  There are suggestions and pointers there that may work for you.