Yes it doesn't matter what you use to toggle the pins, switches are just fine as long as they are debounced as you say.
if I set an arduino pins to input on those control lines do I need to have pull downs?
Not really, but at power up there may be transient pulses on the pins so you may have to do a write to the SR to ensure it is in a known state.
If this is a problem you could also control the OE pin and not allow any hard outputs on the SR until you have initialised it. You could also have pullup/dn resistors on the SR outputs to cause a known state at power up.