One for All SV1630 3 Port HDMI Switcher

Yes, the Smart HDMI switch works with any HDMI cable (connector type). The most recommendable will be the certified HDMI cables that support the required transmission speed. High Speed HDMI cables are recommended.

Almost any HDMI device in the market uses a protocol to show it’s there (HDMI-CEC).
Amazon Fire TV Stick does not support this function.
So it is correct that the automatic function is not supported with Fire TV Stick.

Make sure the power adapter is well connected and the LED on the Smart HDMI Switch is on. Make sure the HDMI cables are properly connected and your sources and TV/display are supporting the right resolution, including the HDCP (High bandwidth digital content protection) see devices into the instruction manual, Wikipedia or manufacturer info on specific make/model.

Connect the source device to the TV/display directly. Does it work correctly?

It might also be the case that (one of) the HDMI inputs does not have enough power to feed the HDMI switch. In that case connect a micro USB cable to the SV1632 to power it. 

Make sure that your DVD, PS3, TV are CEC compatible and enabled on your devices (see instruction manual of your device).