Two points - I was under the impression that the fast idle should begin to operate before the enrichening begins.
Secondly my AEM wideband A/F meter instructions state that running the engine without the sensor connected and powered may lead to failure. I don't think it matters if it has to be turned on and off each time it starts.
Kind of. The sensor usually goes through a heating cycle, but this should be after the engine starts as it lets the exhaust gas blow condensation off the bulb prior to the heating cycle starting. If it heats up then the engine starts, this can cause thermal shock to the bulb and break it.
On my haltech WB1 set up, the ecu won't read or warm the sensor up until the engine has started for a few seconds.
I just wired it into the ignition whilst it was fitted, so it wasn't likely to be left on without the engine running.