Welcome to our KN owner/player group. Glad you have a KN7000. I'm sure it will give you many years of pleasure.
I have a KN6000, but it functions the same as a KN7000 in regard to the questions you raised so I'll try to help you.
Our keyboards have two sounds that are always present. However, they can be changed or silenced. They are ROOT BASS and CHORD. The Fender Rhodes sound you hear is a Modern Electric Piano voice and if you are holding a chord with the left hand when you press the STOP button, you will hear the electric piano sound. To silence the sound, press PROGRAM MENUS/SOUND/PART SETTING and Page 1 of the Part Setting Screen will appear. Press the PART SELECT button (Up Arrow) and hold it down. Scroll to the end. There you will see ROOT BASS. Press the Down Arrow once and you will see CHORD. Set the volume to zero. That silences the Modern E.P. sound. If you want the sound silenced permanently, press SET and CUSTOM PANEL. You can also silence the sound and save your setup in PANEL MEMORY, but would need to save your PANEL MEMORY settings on your SD card or floppy disk.
As for your second question, I know only one answer. To obtain a synchronized STOP, you either have to press the STOP button manually or record your background using one of the three recording methods (Easy Record, Realtime Record, or Step Record), in which case the background will stop automatically at the proper moment.
Hope this helps you.