The phone, to my knowledge, has no way of using the external power before/instead of it's own battery. It works the same way as plugging the phone in or having it on a wireless charger whilst in use. The power management in the phone will shut off the external power supply once the battery is full. It will then keep it topped up on a trickle charge. So the phone will use some battery until it reaches the predetermined level of drain, it will then take the power from the external power source to top it back up.
I don't think so. What is happening when a battery-powered device (phones, laptops) is plugged in and used at the same time, is that the external power source is used to power the device and any surplus power being fed to the device is used to charge the battery*.
Once the battery is at 100% (which is how you would start your day with such a battery pack if you let both charge overnight), the external battery pack is providing the power to operate the device with the internal battery being put on standby mode (ie, is left alone to slowly self-discharge until the self-discharge causes it the fall below some threshold**). Only once the external battery pack is depleted would the device switch to use the internal battery (at which point you could remove the battery case if you want a svelter device).
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* There are situations when the device draws more power than the external source can supply (eg, when charging an iPad with a 5-W charger while heavily using it). Under these circumstances, it will also draw some power from its internal battery and will top up the battery again as the power usage slows down again.
** On Mac laptops that threshold is about 95% and you can clearly see the charge slowly oscillating between 95 and 100% over the course of a day while the laptop is plugged in. On phones that threshold might be closer to 100% but this slow cycling is also pretty much hidden in that the indicator will still show 100% even if the actual charge is only 99 or 98% in order not to unnecessarily worry people.
On iPhones, when the phone is largely sleeping (eg, when in Airplane mode, with the screen off and no audio playing), I've seen no visible self-discharge with the phone still displaying 100% after a 12-h period.