Your power supply is totally undocumented.
I suggest you follow Mike's suggestion first and see how that works out. I would recommend a separate power supply for the pumps and the control system for starters. I'd use a ULN2003A to keep the pumps and the chips happy. You could only use one output at a time if high amps is a concern - the ULN2003a can handle up to 500mA with the right heat sink.
I am also unsure why you need some of the multiplexing chips you're using. Use an analog input then give each switch a different resistor for the Arduino to measure in a voltage divider. One debounce setup, no multiplexer, etc.