1. Are there other settings that affect the keyboard clicks beside the one that I used?
The only way to control the keyboard clicks is by doing the following
1. Turning the silent switch on so you see an orange dot, Located on top left hand side of iPhone
2. Going to Settings - Sounds - Keyboard Clicks - OFF or ON
2. Does the keypad have its own sound setting?
Yes it does, Located in Settings - Sounds - Keyboard Clicks
I would try rebooting your iPhone and see if this continues, Hold down the Home button & Sleep/Wake button together. Once you see a silver apple logo, let go and allow iPhone OS to reboot. Now see if the clicks are doing the same thing.
If your still experiencing the same thing proceed below
Update iPhone to 1.1.4 or if needed Restore iphone and if possible setup as a new iPhone. Please backup all your data before proceeding. Check out the link below to see what programs the iPhone can sync to.
http://support.apple.com/kb/TS1323