1. When the OSSC is turned on, it first shows a grayscale test pattern on the HDMI input as long as it is not synced to a signal on the current input. However, once it has synced to a signal once (SNES turned on) and that signal goes away (SNES turned off after playing), it seems that the HDMI output outputs no signal.
2. I actually asked the designer this question a few days ago, and he said that it should be safe to leave the OSSC turned on. Just mind that it consumes about 3 W even when idling.
3. This goes beyond my expertise, but you should remember that the OSSC requires a power supply that can supply at least 1 A of current at 5 V. You should supply the power from a source that is designed for this.
For a lot of information about the OSSC, please see here.